Mathematics 

OF 

Accounting  and  Finance 


By 

SEYMOUR  WALTON,  A.B.,  C.P.A. 

Dean  of  the  Walton  School  of  Commerce,  Chicago 
And 

H.  A.  FINNEY,  Ph.B.,  C.P.A. 

Professor  of  Accounting,  Northwestern  University 
Chicago 


Third  Printing 


NEW  YORK 

THE  RONALD  PRESS  COMPANY 

1922 


Copyright,  1921,  by 
The  Ronald  Press  Company 


All  Rights  Reserved 


PREFACE 

This  book  has  been  prepared  as  a  manual  of  business  calcu- 
lation, for  use  in  connection  with  the  problems  which  continually 
arise  in  the  course  of  business  as  conducted  today. 

In  deciding  what  subjects  to  include  the  authors  have  been 
guided  throughout  by  the  desire  to  make  the  book  directly  useful 
in  business  offices,  particularly  to  persons  working  in  accounting 
and  in  the  various  lines  of  finance.  There  are  many  school  texts 
on  commercial  arithmetic,  but  they  are  necessarily  too  rudi- 
mentary to  be  of  much  aid  to  readers  of  mature  experience.  No 
space  has  been  given  here  to  a  consideration  of  the  elementary 
and  fundamental  processes  of  mathematics. 

The  effort  has  been,  instead,  to  present  material  of  more 
advanced  nature  which  has  not  been  generally  available.  This 
material  falls  into  three  general  classes. 

The  earlier  chapters  explain  in  considerable  detail  a  number 
of  short  processes  and  practical  suggestions  that  may  be  applied 
in  routine  computations  of  any  sort.  Particular  attention  has 
been  given  to  the  matter  of  adequate  checks  upon  calculations. 

The  central  portion  of  the  book  treats  of  the  special  appli- 
cations of  arithmetical  principles  and  short  methods  to  the 
problems  of  individual  lines  of  business. 

In  the  final  chapters  an  attempt  has  been  made  to  explain, 
in  simple  terms,  convenient  ways  of  using  logarithmic  and  ac- 
tuarial methods  in  the  solution  of  business  problems  relating  to 
compound  interest,  investments,  annuities,  bond  discount  and 
premium,  effective  bond  rates,  leaseholds  and  depreciation. 


During  the  last  months  of  his  life,  Mr.  Walton  devoted  most 
of  his  strength  to  this  book,  and  it  is  a  source  of  profound  personal 


iv  PREFACE 

regret  that  he  did  not  live  to  see  its  publication.  Few  men  were 
his  equal  as  an  accountant,  a  teacher  and  a  writer.  None 
surpassed  him  as  a  friend. 

H.  A.  Finney. 

EvANSTON,  III. 
August  15,  1921 


CONTENTS 

Chapter  Page 

I     Short  Methods  and  Practical  Suggestions     ...        3 

Short  Methods  of  Computation 

Balancing  an  Account 

Combined  Addition  and  Subtraction 

Adding  a  Part  of  a  Column 

Deducting  Several  vSubtrahends  from  one  Minuend 

Table  with  Net  Decrease 

Complements 

Short  Methods  of  Multiplication 

Multiplying  by  1 1 

Multiplying  by  11 1 

Tabulating  Multiples  of  a  Multiplier 

Tabulating  Multiples  of  a  Divisor 

Division  by  Use  of  Reciprocals 

II     Fractions  and  Proof  Figures 16 

Addition  and  Subtraction  of  Fractions 
Cross  Multiplication 

Equivalent  Common  and  Decimal  Fractions 
Proof  Figures 

III     Arithmetical  Progression 22 


Elements  in  Arithmetical  Progression 
Analysis  of  Simple  Progression 
Computing  Total  Simple  Interest 
C.  P.  A.  Problem 

IV     Average 29 

Utility  of  Average 
Simple  Average 
Moving  Average 
Progressive  Average 
Periodic  Average 
Weighted  Average 

V    Averaging  Accounts 42 

Settling  an  Account 
Calculating  Interest 
Items  of  Varying  Amounts 
Focal  Date 
Rules  Applied 


VI  CONTENTS 


Chapter  Page 


Reducing  Days  to  Months 
Compound  Average 
Example  of  Compound  Average 
Another  Illustration 


VI    Percentage 50 

Percentage 

Terms  Used  in  Percentage 

Fundamental  Processes 

Percentage  of  Increase  and  Decrease 

Some  Applications  of  Percentage  in  Business 
Per  Cent  of  Goods  Sold — Various  Manufacturers 
Monthly  Sales  Compared  on  a  Percentage  Basis 
Sales  for  the  Week  Ending  December  18,  1920 
Comparison  of  Sales  by  Departments 
Individual  Sales  Compared  with  Average 
Individual  Sales  Compared  with  Maximum 

Apportionment 
Division  of  Profits 
Distribution  of  Factory  Overhead 

Gross  Profit  Method  of  Approximating  Inventory 

Analysis  of  Statements 

Percentage  Analysis  to  Determine  Causes  of  Variation  in 
Profits 


VII    Equation  in  the  Solution  of  Problems      ....       66 

Solving  Equations 
Illustration  I 
Illustration  2 
Illustration  3 
Illustration  4 


VIII     Trade  and  Cash  Discount  .........       76 

Trade  Discount 

Cumulative  Trade  Discounts 

Methods  of  Finding  Net  Price 

Cash  Discount 

Discount  as  a  Protection  Against  Loss 

Cash  Discount  Regarded  as  an  Expense 

IX    Turnover 83 

Indefinite  Meaning  of  "Turnover" 
Normal  Inventories  Necessary 
Different  Bases  of  Comparison 
Working  Capital  as  Basis  of  Turnover 
De'in'tion  of  Working  Capital 
Need  of  Exact  Definitions 


CONTENTS  vii 

Chapter  Page 

X  Partnerships 89 

Division  of  Profits 

Liquidation  of  Partnerships 

Periodical  Distributions 

Reducing  Capitals  to  Profit  and  Loss  Ratio 

XI    The  Clearing  House 08 


Principle  of  the  Clearing  House 

Debits  and  Credits  with  Clearing  House 

Clearing  House  Transactions 

Manager's  Sheet 

Economy  of  System 

Application  of  Principle  Extended 

XII     Building  and  Loan  Association 103 

General  Characteristics  of  Building  and  Loan  Associations 

Terminating  Plan 

Practicabihty  of  Plan 

Serial  Plan 

Distribution  by  Partnership  Plan 

Distribution  by  Dexter's  Rule 

Withdrawal  of  Shares 

Sources  of  Income 

Premiums 

Individual  Plan 

Dayton  or  Ohio  Plan 

XIII  Good-Will  and  Consolidation 116 

Purchasing  a  Business  with  Stock 

Allocation  of  Net  Earnings 

Good-Will 

Appraising  Good- Will — Year's  Purchase  Method 

Capitalizing  Gross  Income 

Issue  of  Two  Classes  of  Stock 

XIV  Foreign  Exchange 123 

Conversion  of  Foreign  Coinage 
Reverse  Conversion 
Dealing  in  Foreign  Exchange 
Average  Date  of  Current  Account 
Conversion  of  Foreign  Branch  Accounts 
Reconciliation  of  Accounts 

XV    Logarithms i^p 

Use 

Multiplication 

Division 


Vlll 


CONTENTS 


Chapter 


Page 


Calculating  Powers 

Roots 

Nature  of  Logarithms 

The  Characteristic  and  the  Mantissa 

Tables  of  Logarithms 

Characteristics  of  Logarithms  of  Numbers  Between  i  and  lO 

Use  of  the  Characteristic  in  Pointing  off  Results 

Logarithms  of  Numbers  Smaller  Than  i 

Use  of  Negative  Characteristics  in  Pointing  oflF  Answers 

Computing  with  Logarithms  having  Negative  Characteristics 

Determining  Mantissa  by  Interpolation 

Determining  Numbers  by  Interpolation 

XVI    Simple  and  Compound  Interest 154 

Simple  Interest — Methods  of  Calculating 

Rates  Other  Than  6% 

365-day  Basis 

Partial  Payments 

Compound  Interest 

Symbols 

Amount  of  Principal 

Frequency  of  Compounding 

Determining  the  Amount 

Determining  Interest 

Determining  Present  Worth 

Determining  the  Compound  Discount 

Summary 

XVII     Annuities 171 

Definition  of  Annuities 

Symbols 

Amount  of  an  Annuity 

Sinking  Fund  Contribution 

Present  Worth  of  an  Annuity 

Rent  of  an  Annuity 

Equal  Periodical  Payments  on  Principal  and  Interest 

Annuities  Due 

To  Find  the  Amount  of  an  Annuity  Due 

Sinking  Fund 

Required  Annual  Contribution 

Present  Worth  of  an  Annuity  Due 

Rents 


XVIII     Logarithms  in  Compound  Interest  and  Annuity  Com- 
putations     

Calculating  Compound  Interest  and  Annuities  by  Logarithms 
To  Find  the  Compound  Interest  on  i 
To  Find  the  Principal 
To  Find  the  Rate 
To  Find  the  Time 


190 


CONTENTS  IX 

Chapter  Page 

To  Find  the  Present  Value  of  i 

To  Find  the  Compound  Discount  on  i 

To  Find  the  Amount  of  an  Annuity 

To  Find  the  Amount  of  Sinking  Fund  Contributions 

To  Find  the  Present  Worth  of  an  Annuit}' 

XIX     Bond  Discount  and  Premium 194 

Bonds  Purchased  Below  and  Above  Par 

Discount 

Scientific  Method  of  Amortization 

Income  Rates 

Bond  Premium 

Computing  the  Premium  and  the  Price 

First  Method 

Second  Method 

Computing  the  Discount  and  the  Price 

First  Method 

Purchases  at  Intermediate  Date 

Serial  Bonds 

XX    Leaseholds        217 

Commuted  Rents 

XXI     Depreciation  Methods 221 

Annual  Depreciation 

Appendix  A — Values  of  Foreign  Coins 231 

B — Logarithms  of  Numbers 233 

C — Compound  Interest  and  Other  Computations     .     254 


Mathematics  of 
Accounting  and  Finance 


CHAPTER   I 

SHORT  METHODS  AND  PRACTICAL  SUGGESTIONS 

Short  Methods  of  Computation 

Since  the  work  of  an  accountant  necessarily  involves  a  great 
deal  of  computation,  it  is  desirable  that  he  be  familiar  with  cer- 
tain labor-saving  devices.  Those  which  are  presented  in  this 
chapter  have  been  selected  because  of  their  simplicity  and 
because  an  accountant  has  occasion  to  apply  them  so  frequently 
as  to  make  their  use  habitual.  No  attempt  is  made  here  to 
describe  all  the  innumerable  "short  methods"  which,  however 
ingenious,  are  difficult  to  remember  and  rarely  available. 

Balancing  an  Account 

In  striking  a  balance  in  an  account,  the  proper  method  for 
determining  and  inserting  the  balance  is  to  add  the  larger  column 
and  enter  its  total  in  both  columns;  then  to  add  the  smaller 
column  and  insert  each  figure  of  the  balance  necessary  to  produce 
the  total.  The  following  example  is  given  to  illustrate  the 
method. 


Debits 

Credits 

$1,846.22 

$    126.13 

2,913.68 

248.71 

4,327-11 

1,635.48 

2,319.11 

4,757-58 

$9,087.01  $9,087.01 

The  balance  of  $4,757.58  in  this  account  is  found  and  entered 
in  the  following  way.  First,  the  larger  side  of  the  account,  which 
happens  to  be  the  debit  side,  is  added  for  a  total  of  $9,087.01  and 
the  amount  is  placed  on  both  sides;  then  the  items  on  the  credit 

3 


4  MATHEMATICS  OP  ACCOUNTING  AND  FINANCE 

side  are  added  and  the  figure  in  each  digit  column  necessary  to 
give  the  figure  in  the  corresponding  column  of  the  total  is  in- 
serted in  the  place  reserved  for  the  balance. 

Thus,  adding  the  first  column,  3  +  i  +  8  +  i,  gives  13,  to 
which  8  is  added  to  make  21  and  produce  the  i  in  the  total.    The 

8  is  inserted  in  the  balance  as  shown,  while  the  2  of  21  is  carried. 
The  second  column  of  digits  and  the  2  carried  forward  are  added 
to  15,  and  to  this  a  5  is  added  to  make  20  and  give  o  in  the  total. 
The  5  is  entered  in  the  same  column  of  the  balance  and  the  2  of 
20  is  carried.  In  the  same  way  the  third  column  is  added  to  30, 
7  is  inserted  in  the  balance  to  make  37,  and  the  3  of  37  is  carried. 
The  fourth  column  is  added  to  13,  5  is  inserted  to  produce  18,  and 
I  is  carried.  The  fifth  column  is  added  to  13,  7  is  inserted  to 
make  20,  and  2  is  carried.    The  last  column  requires  a  4  to  make 

9  in  the  total,  and  the  entire  balance  is  found  to  be  $4,757.58. 

Combined  Addition  and  Subtraction 

It  often  happens  that  columns  of  figures  are  given  in  which 
are  included  both  positive  and  negative  values,  that  is,  plus  and 
minus  numbers.  It  may  be  necessary,  for  example,  to  compare 
the  figures  of  two  years  to  show  the  increase  or  decrease  of  certain 
items  and  to  determine  the  net  increase  or  decrease  for  the  second 
year.  If  the  figures  are  written  in  ink,  the  minus  quantities  may 
be  inserted  in  red.  In  a  newspaper  or  book  the  minus  numbers 
are  usually  printed  in  italics  or  indicated  by  a  star. 

Two  methods  may  be  followed  in  adding  a  column  of  such 
figures.  The  first  method  is  the  ordinary  way  of  adding  the 
positive  numbers  and  subtracting  from  their  sum  the  sum  of  the 
negative  numbers.  The  other  method  is  to  add  the  numbers 
continuously,  the  negative  digits  being  in  each  case  added  nega- 
tively, that  is  subtracted.  Thus,  in  adding  the  third  column  of 
the  following  table,  which  is  a  comparison  of  the  manufacturing 
costs  of  19 1 6  with  those  of  191 5,  the  procedure  is  as  explained 
below: 


SHORT  METHODS  AND  PRACTICAL  SUGGESTIONS 


Account  1915 

Material  used $12,638.13 

Direct  labor 16,469.42 

Indirect  labor 5,827.59 

Other  manufacturing  expense 7,962.28 


Incre.\se  or 

I9I6 

Decrease 

$14,228.64 

$1,590.51 

19,672.18 

3,202.76 

4,713-92 

1,113.67 

8,242.37 

280.09 

$46,857.11 

$3,959.69 

^2,897•42 


The  net  increase  of  $3,959.69  for  all  four  items  is  found  by 
adding  the  increases  and  decreases  in  the  following  manner.  The 
first  column  of  digits  on  the  right  is  added  thus,  1+6  =  7,  7  —  7 
=  o,  o  +  9  =  9.  The  second  column  is  added  thus,  5  +  7  =  12, 
12  —  6  =  6.  In  the  third  column  3  is  subtracted  from  2  to  make 
—  I,  and  the  i  is  subtracted  from  10  borrowed  from  the  next 
column.  The  addition  is  therefore,  2  —  3  +  10  =  9.  In  the 
fourth  column  the  borrowed  10  is  represented  by  —  i,  and  the 
column  adds  thus,  —1+9  —  1+8  =  15.  The  i  of  15  is 
carried,  as  in  ordinary  addition,  to  the  next  column,  which  added 
is,  1  +  5  +  2  —  1  +  2  =  9.     The  final  column  adds,  i  +  3  — 

I  =  3- 

In  practice,  items  of  the  same  value  but  of  opposite  signs  may 
be  offset  at  once.  Thus,  a  glance  at  the  cents  column  in  the  fore- 
going example  shows  that  i  and  6  offset  —  7,  and  the  remaining 
9  can,  therefore,  be  put  down  at  once  in  the  result. 

Adding  a  Part  of  a  Column 

This  manner  of  addition  is  useful  when  it  is  desired  to  find  the 
sum  of  all  but  the  last  few  numbers  of  a  long  column  whose  total 
is  known.  For  instance,  a  page  may  have  an  amount  on  each  of 
40  lines,  and  the  addition  of  the  entire  column  has  been  verified. 
It  is  now  necessary  to  ascertain  the  total  of  the  first  35  fines,  as  in 
the  illustration  below,  in  which  these  first  35  lines  are  omitted  to 
save  space. 


6  MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 

36th  line $  2,382.46 

37th    "    1,896.28 

38th    "    1,237.61 

39th    " 2,068.42 

40th    "    1,923.16 

Total  of  40  amounts  on  page     $82,642.57 

The  total  of  the  items  to  and  including  the  35th  may  be  found 
by  adding  the  last  five  items  on  a  memorandum  paper  and  deduct- 
ing their  sum  from  the  total  of  the  page.  A  much  quicker  and  more 
workmanlike  procedure,  however,  is  to  subtract  them  by  addition 
as  explained  in  the  previous  section.  Adding  the  first  column  of 
digits,  6  +  8  +  I  +  2  +  6  gives  23.    As  4  is  required  to  make  the 

7  in  the  grand  total,  4  must  be  the  last  digit  in  the  footing  at  the 
35th  hne.  Carrying  2  and  proceeding  in  the  same  way  with 
the  second  and  the  other  columns,  the  required  footing  is  found 
to  be  $73,134.64.  The  entire  operation  involves  no  more 
writing  than  the  insertion  of  the  figures  under  the  35th  line. 

Deducting  Several  Subtrahends  from  One  Minuend 

It  is  sometimes  necessary  to  subtract  several  amounts  from 
one  amount,  as,  for  example,  to  make  various  deductions  from 
total  sales.  If  there  is  room  enough  the  various  items  may  be 
listed  in  an  inside  column,  their  total  inserted  under  gross  sales, 
and  the  difference  carried  out.  If  only  one  column  is  possible, 
the  subtraction  can  be  made  by  addition,  as  shown  in  the  follow- 
ing table,  in  which  the  items  between  the  lines  are  deductions  from 
the  total  sales,  and  the  remainder  is  the  amount  of  the  net  sales. 

Total  sales $132,629.14 

Less: 

Cash  discount $     1,436.27 

Freight 2,389.16 

Breakages 948.63 

Sundry  allowances 769.42 

Net  sales $1 27,085.66 


SHORT  METHODS  AND  PRACTICAL  SUGGESTIONS  7 

In  performing  the  operation  it  should  be  remembered  that  the 
digits  of  the  remainder  are  always  those  necessary  in  each  column 
to  make  the  digit  of  the  same  column  in  the  total. 

Table  with  Net  Decrease 

At  times  the  amounts  to  be  deducted  are  greater  than  the 
amount  from  which  they  are  to  be  deducted.  The  table  below  is 
an  example  of  this.  It  represents  a  comparison  between  two  years 
to  show  the  increaseor  decrease  of  business  in  each  of  seven  depart- 
ments and  the  total  net  result  of  the  whole  business.  The  table 
might  have  been  arranged  in  four  columns,  one  for  the  figures  of 
1920,  one  for  the  figures  of  1921 ,  one  for  the  increases,  and  one  for 
the  decreases;  or  the  departments  might  have  been  listed  in  order, 
with  the  changes  appearing  in  one  column  but  the  decreases 
indicated  by  red  ink  or  italics.  In  the  table  as  given  the  depart- 
ments showing  an  increase  are  listed  first,  and  those  showing  a 
decrease  are  listed  underneath. 

Dept.  1920  1 92 1         Increase  or  Decrease 

I  $168,242.19  $174,629.46  Inc.  $  6,387.27 

3  192,429.36  195,716.83              "  3,287.47 

4  156,283.27  160,142.76             "  3,859-49 
7  84,619.43  86,223.62              "  1,604.19 


Total  $601,574.25  $616,712.67  Tot.  Inc.  $15,138.42 

2  $165,328.46  $158,693.82  Dec.     $  6,634.64 

5  212,642.88  204,976.38  "  7,666.50 

6  198,731.54  189,218.23  "  9.513-31 


Total        $576,702.88        $552,888.43     Tot.  Dec.  $23,814.45 
Grand  total      $1,178,277.13     $1,169,601.10     Net  Dec.    $8,676.03 


The  net  decrease  in  the  foregoing  table  is  the  sum  of  the  de- 
creases less  the  sum  of  the  increases.  The  total  of  the  incieases  is, 
therefore,  added  negatively  to  the  total  of  the  decreases,  that  is, 
deducted  digit  by  digit. 


8  MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 

Complements 

The  arithmetical  complement  of  a  number  is  the  difference 
between  that  number  and  the  next  higher  power  of  lo.  Thus  the 
complement  of  7  is  3,  or  the  difference  between  10  and  7,  and  the 
complement  of  81  is  19,  or  the  difference  between  100  and  81. 

If  instead  of  subtracting  a  number  less  than  10,  its  comple- 
ment is  added,  the  result  is  10  larger  than  the  result  of  the  sub- 
traction. Thus  8  subtracted  from  15  gives  7;  while  the  com- 
plement of  8,  which  is  2,  added  to  15,  gives  17.  Hence  to 
subtract  digits,  in  adding  a  column  of  numbers  negatively, 
their  complements  may  be  added  to  the  other  digits  and  the 
sum  reduced  by  ten  times  the  number  of  complements  added. 

The  following  table  represents  an  addition  of  positive  and 
negative  quantities: 

929 
248 

-  1,086 

-  74 
3,126 

-  174 
306 


3,275 

In  the  unit  column  of  this  table,  9  -]-  8  -f  4  (complement  of  6) 
+  6  (complement  of  4)  +  6  +  6  (complement  of  4)  +  6  =  45,  and 
45  ~"  30  =  15;  30  being  subtracted  because  three  complements 
have  been  added.  The  5  of  the  15  is  entered  in  the  footing  and 
the  I  is  carried.  In  the  tens  column  i  (carried)  +  2  -f  4  +  2 
(complement  of  8)  +3  (complement  of  7)  +  2  -f  3  (complement 
of  7)  =  17.  The  7  is  entered  in  the  footing,  and  since  three 
complements  were  added  the  i  is  dropped  and  —  2  carried  to  the 
hundreds  column.  In  the  hundreds  column  8  (complement  of  the 
—  2  carried)  +  9  +  2-f-i  +  9  (complement  of  i)  +  3  =  32,  and 
32  —  20  (there  being  only  two  complements)  =  12.  The  2  is 
entered  and  the  i  is  carried.    In  the  thousands  column  i  +  9 


SHORT  METHODS  AND  PRACTICAL  SUGGESTIONS  9 

(complement  of  i)  +  3  =  13,  and  13  —  10  (there  being  only  one 
complement)  =  3,  which  is  entered  in  the  footing.  The  result  of 
the  addition  is,  therefore,  3,275. 

Short  Methods  of  Multiplication 

To  multiply  numbers  ending  in  zeros,  both  of  which  are 
integers,  the  significant  figures  of  the  numbers  are  multiplied  and 
to  the  product  are  annexed  as  many  zeros  as  there  are  final  zeros 
in  both  the  multiplicand  and  the  multiplier.  Thus,  3,400  times 
1,200  equals  34  times  12,  or  408,  with  four  zeros  annexed,  or 
4,080,000. 

When  one  of  the  two  numbers  multiplied  is  a  decimal  fraction 
the  significant  figures  in  the  two  numbers  are  multiplied  and  the 
decimal  point  in  the  product  is  moved  as  many  places  to  the  right  as 
there  are  final  zeros  in  the  integer.  This  may  necessitate  annex- 
ing zeros.  In  the  example  .486X300,  .486  is  first  multiplied  by  3, 
to  make  1.458.  As  there  are  two  zeros  in  the  multiplier,  the  deci- 
mal point  in  this  product  is  moved  two  places  to  the  right,  making 
it  145.8. 

To  multiply  by  9,  99,  or  any  number  that  contains  no  other 
figure  than  9,  as  many  zeros  are  annexed  to  the  multiphcand  as 
the  multipher  has  9's,  and  from  the  result  the  multiplicand  is 
deducted.  Thus,  in  multiplying  293  by  99,  two  zeros  are  annexed 
to  293,  making  29,300,  and  from  this  293  is  deducted.  The  result 
is  29,007. 

Multiplying  by  1 1 

To  multiply  an  amount  by  1 1  it  is  not  necessary  to  put  down 
the  figures,  but  only  the  result,  which  is  secured  by  addition. 
The  procedure  is  as  follows : 

Beginning  with  the  right-hand  or  unit  figure  in  the  multipli- 
cand, add  to  each  digit  the  digit  next  to  the  right  of  it.  Put  the 
last  digit  of  this  sum  down  in  the  result,  carrying  the  other  digit 
of  the  sum  if  there  is  one.    Continue  this  until  each  figure  in  the 


10  MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 

multiplicand  has  been  used  twice.    The  method  is  illustrated  in 
the  following  example : 

Illustration 

Multiply  1,342  by  11 

Solution: 

1,342 


Product  14,762 

The  first  digit,  2,  at  the  right  of  the  multiplicand,  has  no 
figure  to  the  right  of  it.  Therefore  nothing  is  added  and  it 
remains  2 

The  second  digit,  4,  is  added  to  the  first  digit,  2,  making  6 

The  third  digit,  3,  is  added  to  the  second  digit,  4,  making  7 

The  fourth  digit,  i,  is  added  to  the  third  digit,  3,  making  4 

Since  the  fourth  digit  has  been  used  only  once  it  is  necessary 
to  use  it  a  second  time,  thus,  i 


The  product  is,  therefore,  14,762 

An  analysis  of  the  ordinary  method  of  multiplication  shows 
that  the  procedure  is  exactly  the  same  in  both  cases.  Looking 
at  the  example  when  multiplied  in  the  usual  way,  it  is  seen  that, 

1342 
II 


1342 
1342 

[4.762 


As  in  the  above  explanation,  the  product  is  formed  by  putting 
down  2  alone,  then  adding  4  to  2,  3  to  4.  i  to  3  and  repeating  the  i. 
It  is,  however,  unnecessary  to  write  out  these  figures,  as  the  whole 
operation  can  be  performed  mentally  and  the  product  put  down 
at  once. 


SHORT  METHODS  AND  PRACTICAL  SUGGESTIONS         1 1 

Multiplying  by  1 1 1 

To  multiply  by  1 1 1  the  same  procedure  is  followed  as  when  the 
multiplier  is  ii,  except  that  each  figure  in  the  multiplicand  is 
added  to  the  two  figures  to  the  right,  and  is,  therefore,  used  three 
times,  as  shown  in  the  following  illustration: 

Illustration 

Multiply  1,342  by  11 1 

Solution: 

1,342 


Product   148,962 

The  first  digit  in  the  multiplicand  is  put  down  alone,  thus,  2 

The  second  digit,  4,  is  added  to  the  first  digit,  2,  making  6 

The  third  digit,  3,  is  added  to  the  second  digit,  4,  and  the 

first  digit,  2,  making  9 

The  fourth  digit,  i,  is  added  to  the  third  digit,  3,  and  the 

second  digit,  4,  making  8 

The  fourth  digit,  i,  is  added  to  the  third  digit,  3,  making  4 

The  fourth  digit,  i,  is  put  down  alone,  thus,  i 


148,962 

If  the  same  method  is  applied  in  multiplying  1,796  by  in,  it 
will  be  found  necessary  to  carry  over  i,  2  and  i,  in  adding  the 
digits. 

In  multiplying  by  any  number  that  consists  of  a  series  of  i  's 
each  digit  in  the  multiplicand  is  used  as  many  times  in  the  way 
explained  above  as  there  are  digits  in  the  multiplier;  twice  when 
the  multipher  is  1 1 ,  three  times  when  it  is  in,  four  times  when  it 
is  1,1 1 1,  and  so  on. 

Tabulating  Multiples  of  a  Multiplier 

When  a  number  containing  several  digits  is  to  be  used  re- 
peatedly as  a  multiplier,  it  saves  time  and  promotes  accuracy  to 
make  a  table  of  its  multiples.    The  best  way  of  compiling  the 


12  MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 

table  is  to  determine  each  multiple  by  adding  the  number  to  the 
preceding  multiple.  The  addition  is,  of  course,  made  on  the  table 
itself  and  not  on  a  separate  sheet  of  paper.  The  table  is  carried 
out  to  ten  multiples  in  order  to  prove  the  work,  as  the  tenth  line 
must  be  the  same  as  the  first  with  one  zero  annexed.  This  is  proof 
only  in  case  the  multiples  are  built  up  by  addition. 

A  table  of  this  sort  constructed  for  683,947  as  a  multiplier  is 
the  following: 


ULTIPLIER 

Multiple 

I 

683,947 

2 

1,367,894 

3 

2,051,841 

4 

2,735,788 

5 

3,419,73s 

6 

4,103,682 

7 

4,787,629 

8 

5,471,576 

9 

6,155,523 

10 

6,839,470 

Supposing  it  is  desired  to  multiply  683,947  by  3,469,  the  solu- 
tion is  as  follows : 

683,947  multiplied  by  9  =  6155523 

"  6  =  4103682 

"         "    4  =  2735788 

"   3  =  2051841 


Product  237261 2143 

Tabulating  Multiples  of  a  Divisor 

When  a  number  containing  several  figures  is  to  be  used  re- 
peatedly as  a  divisor,  a  table  of  multiples  of  the  divisor  may  be 
prepared,  to  show  at  a  glance  how  many  times  the  divisor  is 
contained  in  successive  remainders  and  do  away  with  the  necessity 
of  performing  the  multiplication.  This  method  may  also  save 
considerable  labor  by  reason  of  the  fact  that  instead  of  writing 


SHORT  METHODS  AND  PRACTICAL  SUGGESTIONS         1 3 

down  the  multiples  to  make  each  subtraction,  they  may  be 
written  on  a  card  and  the  particular  multiple  to  be  deducted 
placed  alongside  the  amount  from  which  it  is  to  be  subtracted. 
The  subtraction  is  made  from  the  card  across  to  the  amount,  and 
only  the  remainder  is  set  down. 

The  directions  for  the  use  of  this  method  of  division  may  be 
given  as  follows : 

Placing  the  card  at  the  left  of  the  dividend,  find  by  inspection 
which  multiple  is  to  be  used;  sHde  the  card  until  that  multiple  is 
on  a  line  with  the  dividend;  deduct  the  multiple  from  the  corre- 
sponding first  figures  of  the  dividend,  putting  the  remainder  under 
the  dividend  figures;  then  bring  down  the  next  figure  and  proceed 
in  the  same  way  with  the  new  amount  to  be  divided.  The  follow- 
ing example,  in  which  872,976,654  is  divided  by  683,947,  illus- 
trates this  method. 


Card 

OF  Multiples 

Dividend 

Quotient 

(as  slid  for  each  division) 

I 

683,947 

872976654 

I 

2 

1,367,894 

1890296 

2 

7 

4,787,629 

5224025 

7 

6 

4,103,682 

4363964 

6 

3 

2,051,841 

2602820 

3 

8 

5,471,576 

5509790 
38214 

8 

Quotient  is  1,276.38,  and  the  remainder  382.14. 

Division  by  Use  of  Reciprocals 

Two  numbers  are  reciprocal  when  their  product  is  i.  For 
instance,  2  and  .5  are  reciprocal  numbers,  because  2  X  .5  =  i. 
It  is  evident  from  this  that  the  reciprocal  of  a  number  is  found  by 
dividing  i  by  the  number.  The  quotient  obtained  by  dividing 
by  a  divisor  is  the  same  as  the  product  obtained  by  multiplying 
by  the  reciprocal  of  the  divisor.  For  example,  12  -^  2  and  12 
X  -5  are  each  equal  to  6. 


14  MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 

Repeated  division  by  the  same  divisor  can  be  made  easier, 
particularly  if  an  adding  machine  is  available,  by  determining  the 
reciprocal  of  the  number,  tabulating  the  multiples  of  the  recipro- 
cal, and  proceeding  as  in  multiplication.  In  dividing,  for  example, 
872,976,654  by  683,947,  the  first  step  is  to  divide  i  by  the  divisor 
in  order  to  obtain  its  reciprocal,  which  is  .0000014621.  The 
multiples  of  the  reciprocal  are  then  tabulated,  as  in  the  table 
below.  The  zeros  in  the  reciprocal  are  not  repeated  in  the  table, 
as  it  is  only  necessary  to  remember  the  number  of  places  to  be 
pointed  off  in  the  product,  ten  in  the  present  case.  Having  the 
multipHers,  the  procedure  thenceforth  is  precisely  as  in  multipli- 
cation, as  shown  in  the  second  of  the  subjoined  tables. 


ULTiPLEs  OF  Reciprocal 

I 

14,621 

14621 

2 

29,242 

872976654 

3 

43,863 

58484 

4 

58,484 

73105 

5 

73,105 

87726 

6 

87,726 

87726 

7 

102,347 

102347 

8 

116,968 

131589 

9 

131,589 

29242 

10 

146,210 

102347 
1 16968 

12763791658134 

Pointing  off  ten  decimals,  the  result  is  1,276.38 

Since  the  reciprocal  is  approximate  only,  the  last  figures  of  the 
product  must  be  ignored,  as  they  do  not  show  the  exact 
remainder. 

The  work  may  be  shortened  considerably  by  beginning  at 
the  left  and  ignoring  all  multiples  and  portions  thereof  beyond 
the  first,  second,  or  third  place  to  the  right  of  the  last  place  desired 
in  the  product,  as  follows: 


SHORT  METHODS  AND  PRACTICAL  SUGGESTIONS  15 

14621 
872976654 


8 

I 16968 

7 

102347 

2 

29242 

9 

131589 

7 

10235 

6 

877 

6 

88 

5 

7 

4 

I 

127637917  or  1,276.38 


CHAPTER   II 

FRACTIONS  AND  PROOF  FIGURES 

Addition  and  Subtraction  of  Fractions 

Fractions  having  the  same  numerator  may  be  added  by 
multiplying  the  sum  of  the  denominators  by  the  common  numera- 
tor to  obtain  the  numerator  of  the  result,  and  by  multiplying 
together  the  denominators  of  the  fractions  to  obtain  the  de- 
nominator of  the  result  which  is  then  reduced  to  its  simplest  form. 
To  add  1/9  and  1/7  by  this  method  the  procedure  is  as  follows: 

9  +  7  =  16;  16  X  I  =  16,  the  numerator  of  sum 
9X7  =63,  the  denominator  of  stun 

Sum  =  16/63 

The  solution  of  3/1 1  +  3/16  is  as  follows: 

II  -|-  16  =  27;  27  X  3  =     81,  the  numerator  of  sum 
II  X  16  =  176,  the  denominator  of  sum 

Sum  =  81/176 

To  subtract  fractions  when  the  numerators  are  the  same,  the 
difference  between  the  denominators  is  first  multiplied  by  the 
common  numerator.  This  gives  the  numerator  of  the  result. 
Then  the  denominators  are  multipHed  to  obtain  the  denominator 
of  the  result,  which  is  reduced  to  its  simplest  terms.  This  method 
is  illustrated  in  the  solution  of  the  following  subtraction,  3/1 1  — 
3/16: 

16—  11=515X3=     15,  the  numerator  of  the  difference 
16  X  II  =176,  the  denominator  of  the  difference 

Difference  =  15/176 

Cross  Multiplication 

Two  fractions  may  be  added  by  the  method  of  cross  multipli- 
cation.   The  numerator  of  each  is  multiplied  by  the  denominator 

16 


FRACTIONS  AND  PROOF  FIGURES  l^ 

of  the  other,  and  the  two  products  are  added  to  form  the  numera- 
tor of  the  result.  The  denominators  of  the  fractions  are  multi- 
pHed  to  form  the  denominator  of  the  result,  which  is  then  reduced 
to  its  simplest  terms.  This  method  is  illustrated  in  the  following 
example,  in  which  5/8  and  7/9  are  added. 

5X9=     45 
7X  8=  j6 

loi,  numerator  of  the  sum 
8X9=     72,  denominator  of  the  sum 

Sum  =  101/72  or  I  29/72 

Cross  multiplication  may  also  be  used  in  subtracting  fractions. 
The  numerator  of  each  fraction  is  multiplied  by  the  denominator 
of  the  other,  and  the  smaller  product  is  subtracted  from  the  larger 
to  form  the  numerator  of  the  result.  The  two  denominators  are 
multiplied  to  form  the  denominator  of  the  result.  Reduction  to 
the  simplest  form  then  follows.  In  the  following  example  5/8  is 
subtracted  from  7/9  by  this  method: 

7  X  8  =  56 
5  X  9  =  45 

II,  numerator  of  the  difference 
9X8=  72,  denominator  of  the  difference 
Difference  =  11/72 

The  approximate  product  of  two  mixed  numbers  may  be 
found  by  multiplying  first  the  integers,  then  each  integer  by  the 
other  fraction  to  the  nearest  unit,  and  finally  by  adding  the  three 
products.  The  following  illustration  in  which  1 7  1/5  is  multiphed 
by  14  1/3  shows  the  application  of  the  method : 

14  X  17  =  238 

1/5  of  14  =       3  (nearest  unit) 
i/3  0fi7  =  __6 

Approximate  product  247 
The  exact  product  of  this  multiplication  is  246  8/15. 

If  the  multiplication  of  the  fractions  is  carried  out  to  at  least 
one  decimal  place,  the  result  is  more  exact,  as  the  following  shows: 


i8 


MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 


14X  17  =  238 

1/5  of  14  =       2A 
1/3  of  17=       5.; 


Result  246.5 

Equivalent  Common  and  Decimal  Fractions 

The  following  is  a  list  of  the  most  frequently  used  decimal 
fractions  and  their  equivalent  common  fractions. 


Equivalent 

Decimal 

Common 

Fractions 

Fractions 

5 

y'^ 

■33  1/3 

1/3 

662/3 

2/3 

25 

1/4 

75 

3/4 

16  2/3 

1/6 

^3^/3 

5/6 

125 

1/8 

375 

3/8 

625 

5/8 

87s 

7/8 

081/3 

1/12 

41  2/3 

5/12 

581/3 

7/12 

91  2/3 

11/ 1 2 

0625 

1/16 

1875 

3/16 

3125 

5/16 

4375 

7/16 

5625 

9/16 

6875 

11/16 

8x25 

13/16 

9375 

15/16 

These  figures  can  be  used  in  a  variety  of  ways.  One  of  the 
most  obvious  is  to  substitute  the  common  fraction  for  its  equiva- 
lent decimal,  as  it  is  usually  easier  to  multiply  or  divide  by  the 


FRACTIONS  AND  PROOF  FIGURES  I9 

common  fraction  than  by  the  decimal.  Thus,  if  it  is  desired  to 
find  12  1/2%  of  a  number,  it  is  far  easier  to  divide  the  number  by 
8  than  to  multiply  it  by  the  decimal  .125. 

This  substitution  of  a  common  fraction  for  the  equivalent 
decimal  is  simple  enough  when  the  numerator  of  the  common 
fraction  is  i.  When,  however,  the  numerator  is  greater  than  i, 
it  is  necessary  to  split  up  the  decimal  into  such  parts  as  will  per- 
mit the  substitution  of  equivalent  common  fractions  with  a  nu- 
merator of  I .  The  number  taken  is  multiplied  by  each  of  these 
parts  and  the  products  are  added  to  find  the  result.  The  fewer 
the  parts  into  which  the  decimal  is  split,  the  simpler  is  the  opera- 
tion. The  appHcation  of  the  rule  is  illustrated  in  the  following 
problem. 

Example 

How  much  will  3,648  yards  of  cloth  cost  at  43  3/4  cents  per  yard? 
Solution: 
The  answer  can  be  found  in  three  ways: 

1.  By  multiplying  3,648  by  $.4375. 

2.  By  substituting  for  $.4375  its  equivalent,  7/16  of  $1,  and  mul- 

tiplying 3,648  by  the  latter. 

3.  By  spHtting  up  the  common  fraction  of  7/16  into  1/4,  1/8,  and 

1/16,  or  the  decimal  of  .4375  into  .25,  .12  1/2,  and  .06  1/4,  and 
substituting  the  equivalent  fractions,  and  then  multiplying 
3,648  by  each  of  these  fractions  and  adding  the  products,  as 
shown  below: 

At  $1.00  per  yard  the  price  would  be  $3,648.00 

At  1/4    or  .25  per  yard  the  price  would  be  g  12.00 

At  1/8    or  .12  1/2  (1/2  of  above)  456.00 

At  1/16  or  .06 1/4      "    "       "  228.00 


At  7/16  or  .43  3/4  the  price  is  $1,596.00 


The  operation  can  be  performed  much  more  quickly  if  it  is  recognized 
that  7/16  is  the  same  as  8/16  minus  1/16,  that  is,  1/2  minus  1/8  of  1/2, 
and  the  calculation  is  made  as  shown  below: 


20 


MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 


1/2    of  $3,648  =  $1,824.00 
1/8    of    1,824=        228.00 


7/16  of  $3,648  =  $1,596.00 

It  is  always  advisable  to  prove  all  computations  when  it  does 
not  involve  too  much  work.  This  is  easily  done  in  most  cases 
where  a  fraction  can  be  split  up.  Thus,  in  the  above  example  it 
has  been  found  that  7/16  of  $3,648  is  $1,596.  If  the  figure  for 
1/16  is  added  to  instead  of  subtracted  from  that  for  8/16  the 
result  is  9/16,  and  the  calculation  of  $1,596  can  be  proved  as 
follows  : 


8/16  minus  1/16  is    7/16  or 
8/16  plus      1/16  is    9/16  or 


$1,596.00 

2,052.00 


16/16  or  original  amount  $3,648.00 

Proof  Figures 

There  are  two  systems  of  proof  figures  for  testing  the  accuracy 
of  mathematical  computations.  One  is  based  on  the  casting  out  of 
9's  and  the  other  on  the  casting  out  of  ii's.  Many  bookkeepers 
use  one  or  the  other  system  to  check  all  their  work.  Both  systems 
are  used  in  the  illustration  below  to  check  an  addition,  the  figure 
at  the  right  of  each  amount  being  the  excess  over  the  9's  or  ii's 
in  the  amount,  or  the  remainder  left  after  the  digits  are  added 
and  the  largest  multiple  of  9  or  11  is  subtracted. 


365,592 

286,548 

64,320 

94,094 

810,554 


Excess 

OVER  9's 

3 
6 
6 
8 


_  Total  23,  or  an 
5  excess  of  5 


Excess 

OVER  ii's 

365,592 

7 

286,548 

9 

64,320 

3 

94,094 

°  Total  19,  or  an 

810,554 

8      excess  of  8 

To  cast  out  9's  from  a  number,  the  digits  are  added  ignoring 
9's,  zeros  and  combinations  of  digits  which  add  to  9.    From  the 


FRACTIONS  AND  PROOF  FIGURES  21 

sum  thus  obtained  the  largest  multiple  of  9  contained  therein 
is  subtracted.  Thus,  the  digits  in  the  first  number  in  the  above 
example  are  added  as  follows,  beginning  at  left:  Ignore  the  3  and 
6  because  they  add  to  9;  5  +  5  =  10;  ignore  the  9;  10  +  2  =  12. 
12  —  9  =  3,  th^  excess. 

To  cast  out  1 1 's  begin  with  the  first  figure  at  the  right  and 
add  to  it  the  third,  fifth  and  so  on,  and  take  the  excess  of  their 
sum  over  ii's.  Then  add  together  the  second,  fourth,  sixth 
figures,  etc.,  and  find  the  excess  of  their  sum  over  i  I's.  Subtract 
this  excess  from  the  first  excess  and  the  result  is  the  check  num- 
ber. If  the  first  excess  is  smaller  than  the  second,  add  11  to  it 
before  subtracting.  Thus,  in  the  first  number  of  the  foregoing 
example,  2  +  5  +  6  =  13,  or  an  excess  of  2  over  11.  Similarly, 
9  +  5  +  3  =  17,  or  an  excess  of  6  over  11.  Adding  11  to  the  2 
and  subtracting  6  gives  7  as  the  check  number. 

Addition  is  proved  if  the  excess  over  9's  or  ii's  in  the  sum  of 
the  individual  check  numbers  is  the  same  as  the  check  number 
of  the  total  of  the  numbers  added,  as  shown  in  the  preceding 
example.  This  is  not  an  absolute  proof,  as  an  error  of  9  or  1 1  or 
multiples  thereof  may  be  made. 

Multiplication  is  proved  by  multiplying  the  check  figure  of 
the  multipHcand  by  that  of  the  multiplier.  The  excess  over  9's 
or  ii's  in  the  result  should  be  the  check  figure  of  the  product,  as 
is  seen  from  the  following: 

Excess  Excess 

Numbers  over  9's  overii's 

4,621  4  I 

3,274  7  7 

15,129,154  28  or  I  7 


CHAPTER   III 
ARITHMETICAL  PROGRESSION 

Elements  in  an  Arithmetical  Progression 

An  arithmetical  progression  is  a  series  of  numbers  increasing 
or  decreasing  by  a  common  difference.  The  numbers  in  the  series 
are  called  the  terms;  the  first  and  last  terms  are  called  the 
extremes,  and  the  intermediate  terms  the  means.  An  in- 
creasing or  ascending  series  is  formed  by  adding  the  common 
difference  to  each  preceding  term.  For  example,  7,  12,  17,  22, 
27,  is  an  ascending  series  with  a  common  difference  of  5.  A 
decreasing  or  descending  series  is  formed  by  subtracting  the  com- 
mon difference  from  each  preceding  term.  Thus,  26,  23,  20,  17, 
14,  is  a  descending  series  with  a  common  difference  of  3. 

There  are  five  elements  in  an  arithmetical  progression,  which 
in  the  formulas  to  be  presently  derived  are  represented  by  the 
following  symbols : 

First  term / 

Last  term / 

Common  difference d 

Number  of  terms n 

Sum  of  series 5 

When  any  three  of  these  elements  are  known,  the  other  two 
can  be  computed. 

Analysis  of  Simple  Progression 

The  following  is  an  example  of  a  short  progression: 

ist   term  3 

2nd     "  6 

3d       "  9 

4th      "  12 

Sth      "  15 

6th  or  last  term  18 

r  22 


ARITHMETICAL  PROGRESSION  23 

The  common  difference  here  is  3,  the  number  of  terms  is  6, 
and  the  sum  of  the  terms  is  63. 

It  will  be  seen  that,  although  the  number  of  terms  is  6,  the 
common  difference  is  added  only  five  times.  This  explains  why  in 
Cases  I  and  2  considered  below,  i  is  subtracted,  and  why  in  Case 
3,  I  is  added.  It  will  also  be  seen  that  the  series  consists  of  a 
number  of  pairs,  as  follows: 

ist  and  last  terms  make  a  pair  the  sum  of  which  is  21 
2nd  "  5th  "  "  "  "  "  "  "  "  "  21 
3d      "    4th       "         "     "    "       "      "      "       "     "   21 

The  sum  of  series  is  63 

It  is  evident  from  this  that  the  sum  of  a  series  is  the  sum  of  the 
first  and  last  terms  multiplied  by  half  of  the  number  of  terms. 
It  is  also  evident  that  since  the  sum  of  each  pair  of  terms  is  21, 
the  average  single  term  is  10^,  which  multiplied  by  6,  or  the 
number  of  terms,  gives  63,  or  the  sum  of  the  series. 

If  the  number  of  terms  is  uneven,  the  number  of  pairs  includes 
a  half  pair.  Thus,  if  the  series  given  above  is  extended  to  seven 
terms,  the  last  term  is  21,  and  the  sum  of  the  terms  is  84,  which 
is  3>2  times  24,  the  sum  of  the  first  and  last  terms. 

In  the  following  four  cases  the  derivation  of  the  formulas  for 
computing  the  various  elements  of  an  arithmetical  progression 
is  explained. 

Case  I.    Given  the  first  term,  common  difference  and  the  num- 
ber of  terms,  to  find  the  last  term. 

If  the  series  is  ascending,  the  common  difference  must  be 
added  as  many  times,  less  one,  as  there  are  terms  in  the  series. 
Hence  the  formula  is: 

/  =  /+(«-  i)^ 

Example 

The  first  term  is  9,  the  common  difference  is  3,  and  the  number  of  terms 
is  5.    Find  the  last  term. 


24  MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 

Solution: 

/=    9+(5-  1)3 
=    9+  12 
=  21 

If  the  series  is  descending,  the  common  difference  must  be 
deducted  as  many  times,  less  one,  as  there  are  terms  in  the  series. 
Hence  the  formula  is : 

/  =  /-  („-  i)d 

Example 

The  first  term  is  21,  the  common  difference  is  3,  and  the  number  of 
terms  is  5.    Find  the  last  term. 

Solution: 

/=  21  -  (5  -  i)  3 
=  21—12 
=     9 

Case  2.  Given  the  extremes  and  the  number  of  terms,  to  find 
the  common  difference. 

The  number  of  common  differences  is  one  less  than  the  num- 
ber of  terms;  and  the  sum  of  the  common  differences  is  the  differ- 
ence between  the  extremes.  Hence  in  an  ascending  series  the 
formula  for  finding  the  common  difference  is : 

.=  ^ 

w  —  I 

and  in  a  descending  series  it  is : 


Example 

The  first  term  is  9,  the  last  term  is  21,  and  the  number  of  terms  is  5. 
Find  the  common  difference. 


ARITHMETICAL  PROGRESSION  25 

Solution: 

21  —  Q 
d= 

s-  I 
12 

4 
=    3 

Ca^e  J.  Given  the  extremes  and  the  common  difference,  to 
find  the  number  of  the  terms. 

The  difference  between  the  extremes  is  the  sum  of  the  common 
differences,  and  the  number  of  the  common  differences  is  one  less 
than  the  number  of  terms.  Hence  in  an  ascending  series  the 
formula  for  calculating  the  number  of  terms  is: 

l-f 


d 
and  in  a  descending  series  it  is : 

f-l 


+  I 


+  I 


Example 

The  first  term  is  9,  the  last  term  is  21,  and  the  common  difference  is  3. 
Find  the  number  of  terms: 

Solution: 

21  —  9 


n  = 


12 
=  -+  I 
3 

=  4+1 

=  5 

Case  4.    If  the  extremes  and  the  number  of  terms  are  given, 
the  sum  of  the  terms  is  found  by  the  following  formula : 

s  =  X  n 


26  MATHEMATICS  OP  ACCOUNTING  AND  FINANCE 

or  by  the  following: 


5=    (/+/)X- 
2 


Example 


The  extremes  are  g  and  21,  and  the  number  of  terms  is  5.    Find  the  sum 
of  the  terms. 


Solution: 


9+21  5 

X  5  =  75  or  5  =  (9  +  21)  X  -  =  75 

2  2 


Computing  Total  Simple  Interest 

The  most  important  applications  of  arithmetical  progression 
with  which  accountants  are  concerned  fall  under  Case  4.  They 
are  made  in  computing  the  total  simple  interest  on  a  principal 
which  constantly  increases  or  decreases  from  period  to  period  by 
a  common  difference. 

For  example,  a  $1 ,000,000  bond  issue  bearing  interest  at  5%  is 
to  be  repaid  in  forty  equal  annual  instalments,  and  it  is  necessary 
to  compute  the  total  interest  to  be  paid  during  that  period.  Since 
1/40  of  the  loan  is  to  be  paid  annually,  the  principal  during  the 
last  year  will  be  1/40  of  $1,000,000  or  $25,000.  The  interest  for 
the  first  year  will  be  5%  of  $1,000,000,  or  $50,000,  which  is  the 
first  term  of  an  arithmetical  progression  containing  forty  terms; 
the  interest  for  the  last  year  will  be  5%  of  $25,000,  or  $1,250, 
which  is  the  last  term  of  the  progression.    Since 

2 

the  total  interest,  which  is  s,  will  be 

$50,000  +  $1,250 

X  40,  or  $1,025,000 


ARITHMETICAL  PROGRESSION  27 

C.  P.  A.  Problem 

The  application  of  the  principle  of  arithmetical  progression 
to  the  computation  of  interest  on  a  given  principal  for  an  increas- 
ing or  decreasing  series  of  time  periods,  may  be  illustrated  by  a 
problem  similar  to  one  given  in  an  Illinois  C.  P.  A.  examination. 
The  problem  is  as  follows : 

Problem 

Upon  the  death  of  a  retired  business  man  in  June,  1910,  a  will  is  found 
conveying  real  and  personal  property  aggregating  $300,000  to  the  widow, 
who  is  his  second  wife,  for  her  life,  and  upon  her  death  to  four  children  in 
equal  shares.  It  is  discovered  after  his  death  that  his  first  wife  had  left  to 
her  two  children,  Henry  and  Emma,  $20,000,  consisting  of  securities  for 
$10,000  bearing  6%  interest,  and  uninvested  cash  of  $10,000.  The  father 
had  regularly  collected  the  semi-annual  interest  on  the  investment,  but 
there  was  no  evidence  as  to  his  disposition  of  the  cash  portion  of  the  be- 
quest. Exactly  ten  years  elapsed  between  the  death  of  his  first  wife  and 
his  own  death,  so  that  he  had  collected  twenty  items  of  interest,  the  last 
one  just  before  he  died.  Henry  and  Emma  were  of  age  at  the  time  of 
their  father's  death,  and  had  never  been  informed  of  their  legacy. 

Prepare  a  statement  showing  what  would  accrue  to  each  of  the  four 
children  at  the  death  of  the  widow,  who  died  immediately  after  her  hus- 
band, including  the  amounts  to  which  Henry  and  Emma  would  be  entitled 
on  account  of  their  mother's  estate.  Exclude  and  do  not  consider  any 
accrued  income  of  the  estate  unexpended. 

In  Illinois  the  legal  rate  of  interest  on  undisclosed  debts  is  5%. 

Solution:  The  proceeds  of  the  property  that  belonged  to  his  first 
wife  constitute  a  trust  fund  belonging  to  the  two  children,  Henry  and 
Emma;  the  total  of  this  fund  on  June  i,  1900,  comprises: 

Open  account  for    cash    collected    by  decedent,   June, 

1890 $10,000.00 

Interest  thereon  at  5%  (legal  rate  on  undisclosed  debt) 

for  ten  years 5,000.00 

Principal  of  securities 10,000.00 

Interest  collected  on  securities 6,000.00 

Each  collected  coupon  increased  the  undisclosed  debt  of 
the  father  as  guardian.     The  first  coupon  was  collected 


28  MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 

91/2  years  or  19  half  years  before  the  father's  death.  The 
twentieth  coupon  had  just  been  collected.  The  computa- 
tion of  the  interest  on  these  coupons  can  be  accomplished 
thus: 

First  $300  earns  interest  at  5%  for  9  1/2  years.  .$142.50 
Last  $300     "  "        "     "     "   o  "     . .       o 

Sum  of  extremes $142.50 

Number  of  terms  (coupons) 20 

Applying  the  formula: 

s  =  X  n 

2 

or 

5=  (/+/)X  - 
2 

The  total  interest  =  142.50  X  10  or 1,425.00 

Total  due  Henry  and  Emma  from  mother's  estate.  .  .       $32,425.00 

The  division  of  the  estate  would  be  as  follows: 

Total  real  and  personal  property $300,000.00 

Amount    due    Henry    and    Emma    from    mother's 

estate 32,425.00 

Balance  divided  among  four  children $267,575.00 

Of  which  one-fourth  is $  66,893.75 

The  allowance  of  $1,425  for  interest  on  the  coupons  collected  must  not 
be  confused  with  compound  interest.  The  item  of  $1,425  is  interest  on 
actual  cash  received  by  the  father,  for  which  he  did  not  account.  How  he 
came  into  possession  of  the  cash  is  immaterial. 


CHAPTER  IV 
AVERAGE 

Utility  of  Average 

The  principle  of  average  may  be  used  for  determining  proba- 
bilities, for  comparing  numbers  with  a  standard  or  with  each 
other,  or  for  the  purpose  of  simplifying  calculations  by  using  an 
average  instead  of  a  number  of  related  values.  Where  it  is  pos- 
sible to  collate  statistics  covering  extensive  and  varied  observa- 
tions, it  is  possible  to  determine  an  average  which  may  be 
assumed,  from  the  law  of  averages,  to  be  standard.  Reliance 
can  then  be  placed  on  the  probability  that  other  similar  cases, 
though  individually  at  great  variance  from  the  average,  will  in 
the  aggregate  closely  approximate  the  average.  Mortality 
tables,  for  instance,  are  averages  determined  by  exhaustive 
investigation.  While  they  do  not  determine  probabiHties  for 
individuals,  they  do  determine  probabilities  for  large  groups  of 
individuals. 

When  the  principle  of  average  is  utilized,  the  basis  of  com- 
parison may  be  a  simple  average,  a  moving  average,  a  progressive 
average,  a  periodic  average,  or  a  weighted  average  as  the  case  may 
be.  The  choice  of  a  base  depends  on  the  information  desired  and, 
in  the  case  of  weighted  average,  on  the  necessity  imposed  by  the 
facts  themselves. 

Simple  Average 

The  process  of  determining  a  simple  average  consists  merely  of 
adding  the  units  to  be  averaged  and  dividing  the  sum  by  the 
number  of  units.  If  the  sum  of  the  units  is  known,  the  process 
requires  division  only. 

29 


30  MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 

Suppose  the  daily  sales  for  a  week  are  as  follows : 

Monday.  $3,560.35;  Tuesday,  $3,115.95;  Wednesday,  $2,946.86; 
Thursday,  $2,868.79;  Friday,  $3,269.87;  Saturday,  $3,896.43.  Total 
sales,  $19,658.25. 

The  daily  average  of  the  sales  for  the  week  is  found  by  divid- 
ing the  total  of  $19,658.25  by  the  number  of  days  taken,  or  6, 
which  gives  a  result  of  $3,276.37^. 

It  must  be  remembered  in  figuring  average  that  the  divisor  is 
the  number  of  units  in  the  dividend,  and  not  necessarily  the  num- 
ber of  items  added  to  obtain  the  dividend.  For  example,  suppose 
it  is  desired  to  determine  the  average  contribution  made  by  a 
number  of  persons  to  a  benevolent  fund,  where 


2  men  each  contributed  ' 

I5OO, 

totaling  : 

$1,000.00 

4       " 

a                     (I 

250. 

" 

1,000.00 

10       " 

u                     u 

100, 

" 

1,000.00 

16       " 

a                     li 

50, 

u 

800.00 

18       " 

U                               (( 

25> 

" 

450.00 

so      "     "  "  a  total  of        $4,250.00 

The  number  of  items  added  here  to  find  the  dividend  of  $4,250 
is  only  5,  but  the  number  of  units  to  be  used  as  the  divisor  is  50, 
making  the  average  contribution  equal  to  $4,250  divided  by  50  or 

$85. 

Moving  Average 

When  it  is  desired  to  compare  a  series  of  numbers  relating  to 
units  of  time  of  uniform  duration  and  of  uninterrupted  sequence, 
a  moving  average  of  a  number  of  these  units  may  be  used  as  a 
basis  of  comparison.  This  kind  of  average  is  determined  by  tak- 
ing a  simple  average  of  the  numbers  to  serve  as  the  starting  point 
^or  the  moving  series,  and  dropping  after  the  lapse  of  each  time 
unit  the  first  number  of  the  series  and  adding  the  number  of  the 
next  time  unit  in  order  to  form  a  new  series  and  obtain  a  new 
average. 


AVERAGE  31 

Assume,  for  example,  the  following  conditions : 

1.  The  values  averaged  are  monthly  sales  approximated  to 

the  nearest  $1,000. 

2.  The  time  units  are  months. 

3.  The  number  of  units  is  12,  making  up  one  year. 

4.  The  first  month  of  each  series  is  dropped,  as  a  new  month 

is  included  in  the  next  series. 

In  an  example  of  this  sort  each  month's  sales  can  be  compared 
with  any  one  of  the  twelve  averages  in  the  calculation  of  which 
the  month's  sales  are  included.  For  instance,  the  sales  of  Decem- 
ber, 1914,  can  be  compared  with  the  twelve  monthly  averages  for 
the  years  ending  with  each  month  from  December,  1914,  and  to 
November,  191 5,  inclusive. 

Some  of  the  possible  comparisons  in  which  the  moving  aver- 
age may  be  used  are  illustrated  in  the  table  below.  Their  value 
depends  on  the  nature  of  the  business  whose  figures  are  used. 
The  most  significant  fact  shown  in  the  illustration  is  the  almost 
uninterrupted  increase  in  the  moving  average,  although  the 
monthly  sales  show  wide  variation  due  to  the  seasonal  nature  of 
the  business.  The  decline  in  the  moving  average  during  the  last 
months  of  19 14  and  the  first  months  of  191 5,  shows  the  effects  of 
the  war. 

Following  is  an  explanation  of  the  methods  by  which  the 
figures  in  the  various  columns  are  determined : 

Moving  average: 

Total  sales  for  1 2  months  of  1913 335 

Moving  average  including  December,  1913,  335  -r-  12...  27.  q 

335  —  20  (January,  1913)  +  53  (January,  1914) 368 

Moving  average  including  January,  1914,  368  -^  12  ...  .  30.66 

Increase  or  decrease*  in  moving  average — i  month: 

Moving  average  including  January,  1914 30.7 

"  "  "       December,  1913 27.9 

Increase 2.8 


32  MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 

More  or  less*  than  annual  average: 

December,  1913,  sales 27 

Moving  average  including  December 27.9 

Decrease .9* 

January,  1914,  sales 53 

Moving  average  including  January 30.7 

Increase 22.3 

Increase  in  moving  average — i  year: 

Moving  average  including  December,  1914 35.4 

"  "  "       December,  1913 27.9 

Increase 7.5 

Increase  in  moving  average — 2  years: 

Moving  average  including  January,  1916 48 

"  "  "       January,  1 914 30.7 

Increase 1 7.3 

Moving  Average 


Sales 

Moving 

I9I3 

(000  omitted) 

average 

Jan. 

20 

Feb. 

17 

Mar. 

21 

Apr. 

17 

May 

26 

June 

40 

July 

28 

Aug. 

21 

Sept. 

36 

Oct. 

55 

Nov. 

27 

Dec. 

27 

27.9 

More  or 
less*  than 

annual 
average 


AVERAGE 


33 


Moving  Average —  Continued 


I9I4 

Sales 
(000  omitted) 

Moving 
average 

Increase  or 
decrease*  in 

moving 
average — 
I  month 

More  or 

less*  than 

annual 

average 

Increase 
in  mov- 
ing aver- 
age—  I 
year 

Jan. 

S3 

30.7 

2.8 

22.3 

Feb. 

47 

33-2 

2-5 

13-8 

Mar. 

47 

35-3 

2.1 

II. 7 

Apr. 

45 

37-7 

2.4 

7-3 

May 

?>2 

38.2 

•5 

6.2* 

June 

3,2 

37-5 

•7* 

5.5* 

July 

24 

37-2 

■3* 

13.2* 

Aug. 

17 

36.8 

.4* 

19.8* 

Sept. 

39 

37-1 

•3 

1.9 

Oct. 

42 

36.0 

I.I* 

6.0 

Nov. 

20 

35-4 

.6* 

IS.4* 

Dec. 

27 

35-4 

.0 

8.4* 

7-S 

Increase 


I9IS 

Sales 
(000  omited) 

Moving 
average 

Increase   or 
decrease*  in 

moving 
average — 
I  month 

More  or 

less*    than 

aimual 

average 

or 

decrease* 

in  moving 

average — 

I  year 

Jan. 

45 

34-8 

.6* 

10.2 

4.1 

Feb. 

43 

34-4 

.4* 

8.6 

1.2 

Mar. 

39 

33.8 

.6* 

5-2 

i-S* 

Apr. 

38 

33-2 

.6* 

4.8 

4.5* 

May 

32 

33-2 

.0 

1.2* 

5-0* 

June 

25 

32.6 

.6* 

7.6* 

4.9* 

July 

27 

32.8 

.2* 

5.8* 

4.4* 

Aug. 

32 

34-1 

1-3 

2.1* 

2.7* 

Sept. 

49 

34-9 

.8 

14.1 

2.2* 

Oct. 

85 

38.5 

3-6 

46.5 

2-5 

Nov. 

67 

42.4 

3-9 

24.6 

7.0 

Dec. 

47 

44.1 

1-7 

2.9 

8.7 

34 


MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 


Moving  Average — Continued 


I9I6 

Sales 
(000  omitted) 

Moving 
average 

Increase    or 
decrease*  in 

moving 
average— 
I  month 

More  or 

less*    than 

annual 

average 

Increase 

in 

moving 

average 

—  I  year 

Increase 

in 

moving 

average 

—2  years 

Jan. 

92 

48.0 

3-9 

44.0 

13.2 

^7-3 

Feb. 

97 

52.5 

4 

5 

44-5 

18.I 

19-3 

Mar. 

83 

56.2 

3 

7 

26.8 

22.4 

20.9 

Apr. 

44 

56.7 

5 

12.7* 

23-5 

19.0 

May 

33 

56.8 

I 

23.8* 

23.6 

18.6 

June 

42 

58.2 

I 

4 

16.2* 

25.6 

20.7 

July 

38 

59-1 

9 

21. I* 

26.3 

21.9 

Aug. 

55 

61.0 

I 

9 

6.0* 

26.9 

24.2 

Sept. 

80 

63.6 

2 

6 

16.4 

28.7 

26.5 

Oct. 

97 

64.6 

I 

0 

32.4 

26.1 

28.6 

Nov. 

84 

66.0 

I 

4 

18.0 

23.6 

30.6 

Dec. 

S8 

66.9 

9 

8.9* 

22.8 

3^-5 

Progressive  Average 

Progressive  average  is  cumulative,  a  new  unit  being  added  to 
form  each  successive  dividend,  and  the  divisor  being  constantly 
increased  in  an  arithmetical  progression  of  i .  In  the  table  given 
below  the  figures  used  are  those  of  the  preceding  illustration.  The 
first  average  is  that  of  the  first  two  months'  sales;  the  second,  of 
the  first  three  months'  sales,  etc. 

The  table  shows  the  continuous  growth  of  the  business;  but 
the  differences  between  the  successive  progressive  averages  are 
not  so  significant  as  the  differences  between  the  successive  mov- 
ing averages,  because  the  former  are  borne  down  by  the  smaller 
sales  of  the  first  months  and  years.  Moreover,  an  increase  of  sales 
in  an  early  month  increases  the  progressive  average  for  that 
month  to  a  greater  extent  than  the  same  increase  in  sales  in  a 
later  month  will  increase  the  progessive  average  of  that  month, 
because  the  total  sales  in  the  latter  case  are  divided  by  a  larger 
number  of  months.  The  last  column  of  the  table  indicates  the 
months  in  which  the  sales  run  above  or  below  the  average.    It 


AVERAGE 


35 


will  be  noted  that  the  current  month  is  not  included  in  the  aver- 
age used  as  a  base.  If  it  were  included,  any  increase  or  decrease 
in  the  month's  sales  would  affect  the  standard  as  well  as  the 
month  compared. 

Following  is  an  explanation  of  the  methods  of  determining 
the  figures  appearing  in  the  various  columns : 

Progressive  average: 

February  line     (20  +  1 7)  -J-  2 18.5 

March  line         (20  +  17  +  21)  -r-  3 1Q.3 

Increase  or  decrease"^  in  progressive  average: 

Average  for  first  three  months 19.3 

"         "      "    two  months 18.5 

Increase .8 

More  or  less*  than  progressive  average: 

Sales  of  March,  1913 21 

Progressive  average  for  prior  months 18.5 

Increase 2.5 

Progressive  Average 


Increase  or 

More  or 

DECREASE*    IN 

LESS*  THAN 

Progressive 

PROGRESSIVE 

PROGRESSIVE 

I9I3 

Sales 

AVERAGE 

AVERAGE 

AVERAGE 

January 

20 

February 

17 

18.5 

March 

21 

19-3 

.8 

2-5 

April 

17 

18.8 

.5* 

2.3* 

May 

26 

20.2 

1.4 

7.2 

June 

40 

23-5 

3-3 

19.8 

July 

28 

24.1 

.6 

4-5 

August 

21 

23-8 

•3* 

3-1* 

September 

36 

25.1 

1-3 

12.2 

October 

55 

28.1 

30 

29.9 

November 

27 

28.0 

.1* 

I.I* 

December 

27 

27.9 

.1* 

I.O* 

36 


MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 


Progressive  Average — Continued 


Increase  or 

More  or 

decrease*  in 

less*  than 

Progressive 

progressive 

progressive 

I9I4 

Sales 

average 

average 

average 

January 

S3 

29.8 

1.9 

25-1 

February 

47 

3I-I 

1-3 

17.2 

March 

47 

32.1 

I.O 

15-9 

April 

45 

32.9 

.8 

12.9 

May 

32 

32.9 

.0 

.9* 

June 

32 

32.8 

.1* 

.9* 

July 

24 

32.4 

.4* 

8.4* 

August 

17 

31.6 

.8* 

15-4* 

September 

39 

32.0 

•4 

7-4 

October 

42 

32.4 

•4 

10. 0 

November 

20 

319 

•5* 

12.4* 

December 

27 

31-7 

.2* 

4.9* 

1915 

January 

45 

32-2 

•S 

133 

February 

43 

32.7 

•s 

10.8 

March 

39 

32.9 

.2 

6.3 

April 

38 

33.0 

.1 

5-1 

May 

32 

33-0 

.0 

I.O* 

June 

25 

32.7 

.3* 

8.0* 

July 

27 

32.5 

.2* 

S-7* 

August 

32 

32.S 

.0 

•S* 

September 

49 

330 

•5 

16.S 

October 

85 

34-6 

1.6 

52.0 

November 

67 

35-5 

•9 

32.4 

December 

47 

35.8 

•3 

II-5 

1916 

January 

92 

37-3 

1-5 

56.2 

February 

97 

38.9 

1.6 

59-7 

March 

83 

40.0 

I.I 

44.1 

April 

44 

40.1 

.1 

4.0 

May 

33 

40.0 

.1* 

7.1* 

June 

42 

40.0 

.0 

2.0 

July 

38 

40.0 

.0 

2.0* 

AVERAGE 


37 


Progressive  Average — Continued 


Increase  or 

More  or 

DECREASE*    IN 

LESS*  THAN 

Progressive 

PROGRESSIVE 

PROGRESSIVE 

19 1 6 — Cont. 

Sales 

average 

AVERAGE 

AVERAGE 

August 

55 

40-3 

•3 

15.0 

September 

80 

41.2 

•9 

39-7 

October 

97 

42.4 

1.2 

SS-8 

November 

84 

43-3 

•9 

41.6 

December 

58 

43-6 

•3 

14.7 

Periodic  Average 

In  order  to  show  the  variation  in  the  volume  of  business  be- 
tween seasons,  periodic  average  may  be  utilized,  as  in  the  illus- 
tration below,  in  which  a  simple  average  is  taken  of  the  figures 
for  the  same  month  in  the  years  1913-1916  inclusive: 


Month 

1913 

1914 

1915 

1916 

Total 

Average 

January 

20 

53 

45 

92 

210 

52.5 

February 

17 

47 

43 

97 

204 

51.0 

March 

21 

47 

39 

83 

190 

47-5 

April 

17 

45 

38 

44 

144 

36.0 

May 

26 

32 

32 

33 

123 

30.7s 

June 

40 

32 

25 

42 

139 

34.75 

July 

28 

24 

27 

38 

117 

29.25 

August 

21 

17 

32 

55 

125 

31-25 

September 

36 

39 

49 

80 

204 

5I-0 

October 

55 

42 

85 

97 

279 

69-75 

November 

27 

20 

67 

84 

198 

49-5° 

December 

27 

27 

47 

58 

159 

39-75 

The  principle  of  progressive  average  may  be  utilized  in  con- 
nection with  the  principle  of  periodic  average,  as  indicated  in  the 
following  table : 


i^EAR 

January 

Progressive 
average 

More  or 

LESS*  than   preceding 
PROGRESSIVE    AVERAGE 

I913 

20 

I914 

53 

36.5 

I915 

45 

39-3 

8.5 

I916 

92 

52.5 

52.7 

38  MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 

Each  progressive  average  in  the  foregoing  illustration  is  also 
a  periodic  average.  This  organization  of  the  numbers  and  aver- 
ages makes  possible  a  comparison  of  the  sales  of  each  month  with 
the  average  sales  of  the  same  month  in  all  preceding  years.  The 
figures  in  the  final  column  show  the  difference  between  the  Janu- 
ary sales  of  the  year  and  the  progressive  average  of  the  January 
sales  of  the  preceding  years.  Thus  the  figure  45  for  191 5  is  8.5 
greater  than  the  36.5  progressive  average  for  January,  191 3,  and 
1914. 

Weighted  Average 

When  the  values  entering  into  the  computation  of  an  average 
differ  in  two  or  more  particulars,  a  simple  average  is  impossible. 
Take  the  following  case,  for  example : 

3  men  earn  $5.00  per  day 
S     "        "       6.00     "      " 

4  "        "       7-00     "       " 

In  computing  the  average  daily  wage  of  these  twelve  men, 
the  fact  must  be  recognized  that  the  wage  payments  differ  in 
two  particulars: 

1 .  The  daily  wage 

2.  The  number  of  men  receiving  each  wage 

The  daily  average  is  found  by  dividing  the  aggregate  of  their 
daily  wages  by  their  number,  as  shown  below : 


Men 

Wage 

Product 

3 

$5 

$15 

5 

6 

30 

4 

7 

28 

12  $73 

$73  -^  12  =  $6Vi2,  the  average  daily  wage 

This  example  serves  to  illustrate  the  principle  that  each  value 
must  be  weighted  by  multiplying  it  by  the  number  of  units  to 


AVERAGE  $9 

which  the  value  is  applicable.    Additional  illustrations  will  make 
the  principle  clearer. 

Illustration  i 

What  is  the  average  rate  of  interest  earned  on  the  following  investments 
made  for  one  year? 


$100  at 

7% 

$5,000  at 

6% 

$15,000  at 

5% 

Solution: 

Principal 

Rate 

Product 

$         lOO 

7% 

$        7.00 

5,000 

6% 

300.00 

15,000 

5% 

750.00 

$20,100 

$1,057.00 

i)057  ^  20,100  =  5.258%,  the  average  rate  of  interest 


Illustration  2 

What  is  the  average  life  of  a  plant,  the  various  fixed  assets  of  which 
have  the  following  costs  and  estimated  lives? 


( 

Class  of 

Asset 

Cost 

Life 

A 

$  3,000.00 

5  years 

B 

15,000.00 

10     " 

C 

35.000.00 

20     " 

Solution: 

Depreclation 

Annual 

Class 

Cost 

Life           Rate 

D 

EPRECIATIC 

A 

$  3,000.00 

5  years           20% 

$    600.00 

B 

15,000 

.00 

10      "               10% 

1,500.00 

C 

35,000, 

.00 

-^0     •'                5% 

1,750.00 

$53,000. 

00 

$3,850.00 

53,000-^  3.850=  13.76+  ^o  the  average  life 


40 


MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 


Illustration  3 

What  is  the  average  per  cent  of  increase  in  the  cost  of  manufacture 
under  the  following  conditions:  Where  one  pound  of  each  item  is  used  in 
the  manufacture  of  each  completed  article,  and  the  cost  of  each  iiem  has 
increased  by  the  per  cent  shown  in  the  last  column  of  the  first  of  the  sub- 
joined tables? 

Material      Cost  per  Pound     Per  cent  Increase 


1917 

918 

A 

$4 

200 

B 

5 

10 

C 

8 

6X 

D 

IS 

10 

E 

18 

20 

Solution  i  : 

No. 

Cost  per 

Per  cent 

Material 

Pounds 

Pound 

Increase     I 

NCREASE 

.1917 

1918 

1918 

A 

$  4 

200 

$8.00 

B 

5 

10 

•50 

C 

8 

ty^ 

•50 

D 

IS 

10 

1.50 

E 

18 

20 

3.60 

$14.10 


14.10   -^  50  =  28.2%,  average  increase 


Solution  2: 

Cost  per 

Per  cent 

Per  cent 

Weighted 

Material 

Pound 

OF 

Total 

Increase 

Per  cent 

1917 

1918 

1918 

A 

$4 

8 

200 

16.0 

B 

S 

10 

10 

I.O 

C 

8 

16 

6K 

I.O 

D 

IS 

30 

10 

30 

E 

18 

36 

20 

7.2 

$50 


28.2 


AVERAGE 


41 


Illustration  4 

The  same  conditions  are  taken  here  as  in  the  preceding  illustration, 
except  that  the  items  composing  the  finished  article  are  each  of  a  different 
weight,  as  indicated  in  the  second  column  of  the  following  two  solutions: 


Solution 

i: 

No. 

% 

Cost  per 

Weighted 

% 

Weighted 

Material 

Pounds 

Total 

Pound 

Product 

Increase 

% 

1917 

1917 

1918 

1918 

A 

2 

10 

$  4 

$  40 

200 

80 

B 

8 

40 

5 

200 

10 

20 

C 

4 

20 

8 

160 

(^Va 

10 

D 

4 

20 

15 

300 

10 

30 

E 

2 

10 

iS 

180 

20 

36 

20  100  fooo 

176  -^  880  =   20%,  weighted  average  percent 


176 


Solution  2: 

No. 

Cost  per 

Total 

% 

Material 

Pounds 

Pound 

Cost 

Increase 

Increase 

1917 

1917 

1918 

1918 

A 

2 

$  4 

$     8 

200 

$16.00 

B 

8 

5 

40 

10 

4.00 

C 

4 

8 

^2 

6>^ 

2.00 

D 

4 

15 

60 

10 

6.00 

E 

2 

18 

36 

20 

7.20 

$176 


$35-20 


35.20  -j-  176  =  20%,  weighted  average  per  cent 


CHAPTER  V 
AVERAGING  ACCOUNTS 

Settling  an  Account 

The  object  of  averaging  an  account  is  to  determine  a  single 
date,  known  as  the  average  date,  on  which  the  account  may  be 
settled  with  fairness  to  both  debtor  and  creditor. 

As  a  simple  illustration  of  how  an  account  is  averaged,  sup- 
pose B's  account  with  A,  to  whom  he  is  indebted,  is  as  follows: 

March    i — 60  days $1,000 

March  31 — 60  days 1,000 

The  first  item  of  this  account  is  due  on  April  30  and  the  second 
on  May  30.  B  may  pay  each  amount  at  its  maturity,  or  the  entire 
$2,000  at  the  average  maturity,  which  is  May  15.  He  can  make 
an  equitable  settlement  on  this  average  date  because  the  time  he 
gains  in  deferring  payment  of  the  first  $1,000  for  fifteen  days  is 
exactly  offset  by  the  time  he  loses  in  paying  the  second  $1,000 
the  same  number  of  days  in  advance  of  the  due  date. 

It  may  be  desirable  to  compute  the  average  date  for  the 
purpose  of  dating  or  determining  the  maturity  of  a  note  or  other 
document  given  in  settlement  of  an  account.  For  instance,  B 
might  cover  his  account  by  giving  A  a  single  non-interest  bearing 
note  for  $2,000,  due  May  15,  instead  of  two  non-interest  bearing 
notes  of  $1,000  each,  one  due  April  30,  and  the  other  due  May  30. 

Calculating  Interest 

The  average  date  may  also  be  desired  for  the  purpose  of  com- 
puting interest  on  the  balance  instead  of  the  individual  items  of  an 
account.  For  instance,  if  A  should  settle  the  account  on  June  14 
by  a  single  payment  of  $2,000,  equity  would  require  that  he  pay 
interest,  say,  at  6%  as  follows: 

42 


AVERAGING  ACCOUNTS  43 

On  $i,ooo  from  April  30  to  June  14 — 45  days $  7.50 

On  $1,000  from  May  30  to  June  14 — 15  days 2.50 

Total $10.00 

The  interest  might,  however,  be  computed  on  the  entire  bal- 
ance of  $2,000  for  a  period  of  thirty  days  from  the  average  date, 
May  15  to  June  14.    Its  amount  in  this  case  would  also  be  $10. 

Assume  that  instead  of  paying  cash,  B,  on  April  14  gave  A  a 
note  for  $2,000  due  in  two  months.  The  maturity  of  this  note 
would  be  June  14,  or  thirty  days  after  the  average  date.  An 
equitable  settlement  would  require  the  addition  of  $10  interest 
to  the  face  of  the  note,  which  would  make  it  $2,010. 

Items  of  Varying  Amounts 

In  the  foregoing  illustrations  the  amounts  are  the  same  and 
only  the  varying  number  of  days  has  had  to  be  considered  in 
arriving  at  the  average  date.  If,  however,  the  amounts  are  not 
the  same,  they  must  also  be  considered.  Suppose,  for  example, 
that  B's  account  with  A  was  as  follows: 

March  i — 60  days $2,000 

"     31 — 60  days 1,000 

The  $2,000  item  is  due  April  30  and  the  $1,000  item  is  due  May 
30.  In  this  case  the  average  date  of  maturity  for  the  total  of 
$3,000  would  not  be  May  15  as  in  the  previous  example,  since  the 
payment  of  $2,000  fifteen  days  after  maturity  would  not  be  off- 
set by  the  payment  of  $1,000  fifteen  days  before  maturity.  It 
would  be  May  10,  because  $2,000  paid  ten  days  after  maturity 
would  be  counterbalanced  by  $1,000  paid  twenty  days  before 
maturity.  It  is  evident  from  this  that  in  averaging  accounts 
due  consideration  must  be  given  to  amounts  as  well  as  to  dates. 
The  dates  to  be  used  in  averaging  an  account  are  those  at 
which  the  items  may  be  assumed  to  have  a  cash  value  equivalent 
to  the  amount  at  which  they  appear  in  the  accounts.  Thus,  sales 
on  cash  terms,  being  due  on  the  day  of  sale,  take  their  invoice 


44  MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 

dates  in  averaging;  sales  with  credit  terms  take  the  dates  on 
which  the  invoices  are  due;  returns  and  allowances  take  the  dates 
when  the  invoices  to  which  they  apply  are  due;  an  interest-bear- 
ing note  takes  the  date  of  the  note  since  the  face  is  the  cash  value 
at  that  date;  a  non-interest  bearing  note  is  not  worth  its  face 
until  due  and  hence  its  maturity  is  used  in  the  calculation  of  the 
average  date. 

Focal  Date 

The  average  dates  in  the  preceding  illustrations  were  deter- 
mined by  inspection.  When  an  account  is  not  so  simple  the  com- 
putation of  its  average  date  requires  a  method  involving  the 
principle  of  weighted  average  and  the  selection  of  a  basic  date  for 
calculating  the  time  of  each  item.  This  basic  date  is  called  the 
focal  date  and  the  one  most  advantageously  employed  is  the  last 
day  of  the  month  preceding  the  earliest  date  used  in  averaging. 

Take,  for  example,  the  following  account: 

Date  of  Terms  of        Date  Used 

Transaction         Payment      in  Averaging      Amount 

March  i  60  days  April  30  $2,000.00 

"    31  60      "  May  30  1,000.00 

The  first  date  to  be  used  in  averaging  the  account  is  April  30. 
Hence  March  31,  the  last  day  of  the  preceding  month,  is  selected 
for  the  focal  date.  The  subsequent  steps  in  the  process  of  averag- 
ing the  account  may  be  enumerated  as  follows: 

1.  Assume  that  each  item  is  paid  on  the  focal  date. 

2.  Determine  the  number  of  days  each  item  would  be  pre- 

paid if  it  were  paid  on  the  focal  date. 

3.  Multiply  its  amount  by  this  number  of  days. 

4.  Add  the  products  thus  obtained. 

5.  Divide  this  sum  by  the  total  of  the  items;  the  quotient 

represents  the  number  of  days  the  focal  date  precedes 
the  average  date. 


AVERAGING  ACCOUNTS  45 

Rules  Applied 

The  application  of  these  rules  to  the  foregoing  account  is  as 
follows : 

1.  The  two  items  are  assumed  to  be  paid  on  March  31,  the 

focal  date  taken. 

2.  The  first  item  is  therefore  assumed  to  be  prepaid  30  days, 

and  the  second  60  days. 

3.  Paying  $2,000  30  days  before  it  is  due  is  equivalent  to 

paying  $1,  60,000  (2,000  X  30)  days  before  it  is  due; 
and  paying  $1 ,000  60  days  before  it  is  due  is  equivalent 
to  paying  $1 ,  60,000  (i  ,000  X  60)  days  before  it  is  due. 

4.  Hence  the  two  assumed  prepayments  are  equivalent  to  a 

prepayment  of  $1  by  120,000  days. 

5.  One  dollar  prepaid  120,000  days  is  equivalent  to  $3,000 

prepaid  1/3 ,000  of  1 20,000  days,  or  40  days.    The  aver- 
age date  is  therefore  40  days  forward  from  March  31. 
Forty  days  are  taken  as  the  equivalent  of  i  month  and 
10  days,  making  the  average  date  May  10. 
The  computation  of  the  average  date  is  shown  in  tabular  form 

as  follows : 

Date  of       Terms  of    Date  in    Time   from   Focal 
Transaction    Payment     Average    to  Maturity  Date  Amount  Product 

March    i         60  days       April  30  30  days  $2,000      $60,000 

March  31         60    "  May  30  60    "  1,000       60,000 

$3,000    $120,000 

Dividing  the  sum  of  the  products  by  the  sum  of  the  amounts 
(120,000  -^  3,000)  gives  40,  or  the  number  of  days  the  average 
date  follows  the  focal  date. 

Reducing  Days  to  Months 

When  the  time  between  the  focal  date  and  the  date  of  any  of 
the  items  in  the  account  is  more  than  one  month,  the  reduction 
of  this  time  to  days  may  be  avoided  by  the  method  outlined 
below : 


46  MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 

1 .  Express  the  time  in  months  and  days. 

2.  Multiply  the  amount  of  each  item  by  the  number  of 

months  to  obtain  a  product  of  months;  and  by  the 
number  of  days  to  obtain  a  product  of  days. 

3.  Add  the  products  of  months;  also  the  products  of  days. 

4.  Reduce  the  sum  of  the  products  of  months  to  days  by 

multiplying  by  30. 

5.  To  the  product  of  days  thus  obtained,  add  the  product  of 

days  previously  obtained. 

6.  Divide  this  sum  by  the  balance  of  the  account  to  find 

the  time  in  days  between  the  focal  and  the  average 
dates. 

7.  Reduce  this  time  to  months  and  days  on  the  basis  of 

thirty  days  to  a  month. 

This  method  is  illustrated  in  the  following  example,  in  which 
the  focal  date  taken  is  February  28,  being  the  last  day  of  the 
month  preceding  the  earliest  date  in  the  average,  which  is 
March  3. 


Date  of 

Terms  of 

Date   in 

Time 

Product 

Transaction 

Payment 

Average 

Mos 

1.    Days  Amount  Mos.     Days 

March    3 

cash 

March    3 

0 

3 

$250    $       0    $    750 

March  18 

I  month 

April     18 

I 

18 

500        500      9,000 

April     10 

30  days 

May      10 

2 

10 

200        400      2,000 

May       8 

cash 

May       8 

2 

8 

400        800      3,200 

$1,350  $1,700  $14,950 
30  X  1,700=51,000 

$65,950 

65,950  -T-  1,350  =48  115/135.  or  49  days 
49  days  =  I  month  and  19  days 

One  month  and  nineteen  days  forward  from  February  28,  the 
focal  date,  is  April  19,  which  is  the  average  date. 

This  computation  is  based  on  the  assumption  that  there  are 
30  days  in  each  month  and  360  days  in  a  year.  Strictly  speaking, 
however,  the  item  due  May  10  runs  for  seventy-one  days  instead 


AVERAGING  ACCOUNTS  47 

of  seventy,  and  the  item  due  May  8  runs  for  sixty-nine  days  in- 
stead of  sixty-eight.  The  result,  however,  would  not  be  ma- 
terially modified  if  the  exact  number  of  days  were  taken.  The 
other  method  is,  therefore,  sufficiently  accurate  for  all  ordinary 
commercial  transactions. 

Compound  Average 

The  average  date  of  an  account  is  computed  by  simple  aver- 
age, when  the  account  contains  either  debit  or  credit  items,  but 
not  both.  When  both  debits  and  credits  are  included,  the  aver- 
age date  is  found  by  means  of  compound  average,  which  involves 
the  following  steps: 

1.  Determine  the  products  of  months  and  days  for  the  debits 

and  credits. 

2.  Determine  the  diff'erence  between  the  sum  of  the  debit 

products  and  the  sum  of  the  credit  products. 

3.  Divide  this  difference  by  the  balance  of  the  account. 

4.  If  the  difference  of  the  products  is  on  the  same  side  of  the 

account  as  its  balance,  the  average  date  is  forward 
from  the  focal  date;  but  if  the  difference  of  the  products 
and  the  balance  of  the  account  are  on  different  sides, 
the  average  date  is  backward  from  the  focal  date.  This 
latter  condition  rarely  occurs  if  the  focal  date  selected 
is  prior  to  the  dates  of  the  items  in  the  account. 

Example  of  Compound  Average 

In  illustrating  compound  averaging  the  following  account  is 
taken : 

Debit  Credit 

June        I $500    July         5  Note  (2  mo.  with- 

"         20,  I  mo 400  out  int.) $500 

July       10 600         "         10  Returns  (Inv. 

August    5 500  June  20) 50 

August    I  Cash 300 


48  MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 

As  the  July  5  th  note  on  the  credit  side  of  the  account  is  non- 
interest  bearing,  it  does  not  have  a  cash  value  of  $500  until  its 
maturity  on  September  5.  It  therefore  takes  this  date  in  the 
average.  The  July  loth  credit,  being  an  offset  to  the  debit  of 
June  20,  takes  the  same  maturity  date  in  the  average  as  the  debit 
item,  or  July  20th.  As  June  i  is  the  earliest  date  on  which  any  of 
the  items  in  the  account  has  a  cash  value  equal  to  the  face  of 
the  item,  the  most  convenient  focal  date  is  May  31. 

The  sum  of  the  products  of  months  and  days  for  the  debit 
items  of  the  account  is  found  as  follows : 

n 
Date  of       Terms  of    Date  in 
Transaction    Payment    Average     Mos. 

June        I  June        i  o 

"        20  I  month     July       20  i 

July       10  July       10  I 

August    5  August    5  2 

Total  debits $2,000  $2,000  $17,000 

30  X  2,000  =     60,000 

Sum  of  the  debit  products $77,000 

The  sum  of  the  products  of  months  and  days  for  the  credit 
items  is  found  by  the  following  computation. 

Date  of       Terms  of      Date  in  ^ime  Products 

Transaction    Payment      Average  Mos.      Days  Amount  Mos.     Days 

^500    $1,500  $  2,500 

SO  50        1,000 

300        600  300 


T 

Products 

)ays 

Amount  Mos.     Days 

I 

$500  $       0  $      500 

20 

400        400       8,000 

10 

600        600       6,000 

5 

500     1,000       2,500 

July      5 

2  months     Sept.       5 

3 

5 

July     10 

offset 

June    20 

Dr.               July      20 

I 

20 

August    I 

August    I 

2 

I 

Total  credits 

50  $2,150  $  3,800 
2,150  X  30  =  64,500 

Sum  of  the  credit  products $68,300 

The  account  has  a  debit  balance  of  $1,150  and  the  sum  of  its 
debit  products  exceeds  the  sum  of  its  credit  products  by  8,700. 


AVERAGING  ACCOUNTS  49 

The  average  date  is  forward  from  May  3 1 ,  the  focal  date,  as  many 
days  as  the  number  of  times  1,150  is  contained  in  8,700,  or  ap- 
proximately 8.  June  8  is,  therefore,  the  focal  date.  The  account 
could  be  averaged  by  deducting  the  $50  credit  from  the  June  20th 
debit  of  $400  and  dealing  only  with  the  net  debit  of  $350. 

Another  Illustration 

To  illustrate  the  conditions  under  which  the  average  date  is 
backward  from  the  focal  date,  assume  that  the  July  5th  non- 
interest  bearing  note  was  due  in  six  months.  All  of  the  debit 
items  are  due  on  or  before  August  5,  but  since  the  creditor  would 
have  to  wait  until  January  5  for  the  $500  payable  on  the  note, 
which  in  the  meantime  would  earn  no  interest,  the  balance  of  the 
account  should  carry  an  early  average  maturity.  The  sum  of  the 
debit  products  of  days  would  be  computed  as  in  the  preceding 
example  and  would  total  77,000,  while  the  sum  of  the  credit  pro- 
ducts of  days  would  be  computed  as  follows: 


Date  of       Terms  of    Date  in 


Time  Product 


Trans.\ction    Payment  Average     Mos.  Days  Amount  Mos.  Days 

July         5        6  months  January    57  5       $500    $3,500  $  2,500 

July       10       offset  July         20      i  20           50           50  1,000 

August    I  August      12  I         300         600  300 

Total  credits $850    $4,150  $    3,800 

30  X  4,150  =   124,500 
Sum  of  the  credit  products $128,300 

The  balance  of  the  account  would  still  be  a  debit  of  $1,150, 
but  the  difference  of  the  products  of  days  would  now  be  on  the 
credit  side,  and  would  amount  to  51,300.  The  average  date 
would,  therefore,  be  backward  from  June  i,  the  focal  date,  by 
forty-five  days,  or  the  number  of  times  $1,150  is  contained  in 
51,300.  Counting  thirty  days  to  the  month,  the  average  date 
would  be  one  month  and  fifteen  days  backward  from  the  focal 
date  and  would,  therefore,  be  April  15. 


CHAPTER  VI 
PERCENTAGE 

Percentage 

Percentage  is  a  method  of  computing  by  hundredths.  The 
symbol  %  means  per  cent  or  hundredths.  A  rate  per  cent  is 
equivalent  to  a  common  fraction  the  numerator  of  which  is  ex- 
pressed and  the  denominator  of  which  is  indicated  by  the  symbol 
%  as  being  loo.  Thus  the  same  facts  may  be  stated  in  the  form 
of  a  common  fraction,  a  decimal  fraction  or  a  per  cent.  The 
following  are  equivalent: 

Common  Fractions  Decim.\l  Fractions  % 

17/100  .17  17 

iX  1. 25  125 

iH  7-So  750 

12/4  3.00  300 

Terms  Used  in  Percentage 

Base.  The  number  of  which  a  given  per  cent  is  to  be  taken  is 
called  the  base. 

Rate.    The  per  cent  of  the  base  to  be  taken  is  called  the  rate. 

Percentage.  The  result  obtained  by  taking  a  certain  per  cent 
of  the  base  is  called  the  percentage. 

Fundamental  Processes 

All  mathematical  computations  involving  percentage  may  be 
grouped  under  three  headings : 

I.  To  find  a  given  per  cent  of  a  number;  that  is,  to  find  the 
percentage. 

Rule:     Multiply  the  base  by  the  rate 
Example:  $60    X  20%  =  $12 

Base  X  Rate  =  Percentage 

50 


PERCENTAGE  5 1 

2.  To  find  what  per  cent  one  number  is  of  another;  that  is,  to 

find  the  rate. 

Rule:     Divide  the  percentage  by  the  base 
Example:  $12         -r-  $60    =  20% 

Percentage  -H  Base  =  Rate 

3.  To  find  a  number  when  a  certain  per  cent  of  it  is  known; 

that  is,  to  find  the  base. 

Rule:     Divide  the  percentage  by  the  rate 
Example:  $12         -7-  20%  =  $60 

Percentage  -^  Rate  =  Base 

Percentage  of  Increase  and  Decrease 

Percentage  is  frequently  employed  to  compare  numbers  and 
to  show  how  much  larger  or  smaller  one  number  is  than  the  other. 
No  new  mathematical  principles  are  involved  in  such  computa- 
tions, as  may  be  shown  by  the  following  illustrations: 

1 .  Percentage  of  Increase.  The  sales  of  a  certain  business  in 
May,  1919,  were  $16,000  while  the  sales  in  May,  1920,  were 
$18,000. 

The  smaller  number  is  taken  as  the  base;  the  difference  be- 
tween the  two  numbers  is  the  percentage  of  increase.    Then 

2,000  -f-    16,000   =  12^% 

Percentage  of  increase  -^  Base     =  Per  cent  of  increase 

2.  Percentage  of  Decrease.  The  profits  of  a  business  for  the 
year  1919  were  $20,000  while  the  profits  for  1920  were  $17,000. 

The  larger  number  is  taken  as  the  base;  the  difference  be- 
tween the  two  numbers  is  the  percentage  of  decrease.    Then 

3,000  -7-  20,000  =  15% 

Percentage  of  decrease  4-    Base    =  Per  cent  of  decrease 

Some  Applications  of  Percentage  in  Business 

Comparisons 

Business  statistics  may  be  tabulated  and  compared  on  a  per- 
centage basis  to  determine  the  relative  effectiveness,  desirabihty 


Number 

Per  Cent 

Sold 

Sold 

293 

79.6 

316 

64.1 

582 

91. 1 

416 

79-5 

52  MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 

or  productivity  of  similar  factors.    The  illustrations  given  in  this 
chapter  are  intended  to  be  suggestive  but  not  exhaustive. 

The  following  tabulation  shows  the  number  of  units  of  a  cer- 
tain commodity  purchased  from  various  manufacturers  during  a 
year,  the  number  sold,  and  the  per  cent  sold.  A  comparison  of 
the  per  cents  indicates  the  relative  salability  of  the  goods  pur- 
chased from  the  various  manufacturers. 

Per  Cent  of  Goods  Sold — Various  Manufacturers 

Number 
Manufacturer  Purchased 

Walker  &  Co 368 

White  &  Dudley 493 

Davis  Mfg.  Co 639 

Barton-Walsh 523 

The  following  tabulation  compares  the  sales  of  various  sales- 
men during  a  month.  Each  man's  sales  (as  a  percentage)  divided 
by  the  total  sales  (as  a  base)  produces  a  rate  which  measures  his 
portion  of  the  total. 

Monthly  Sales  Compared  on  a  Percentage  Basis 

Name  Sales            Per  Cent  of  Total 

Arthur  Bradley $  1,264.90                        7.8 

J.B.Henderson 1,913.52  11. 8 

Fred  Bates 1,732.69  10.6 

Arthur  Dutton 2,213.72  13.6 

J.  L.  Weston 1,963.45  12.1 

Carter  Doane 1,627.32  lo.o 

Walter  S.  Waite 1,692.18  10.4 

Frank  Chesley 2,138.45  13. i 

Harold  Peters 1,728.46  10.6 

Total $16,274.69  loo.o 


The  statistics  may  be  so  arranged  as  to  obtain  two  percentage 
analyses,  as  illustrated  in  the  following  tabulation  which  shows 


PERCENTAGE 


53 


what  per  cent  of  the  sales  of  the  week  was  made  by  each  salesman, 
and  what  per  cent  was  made  each  day. 

Sales  for  the  Week  Ending  December  i8,  1920 


D\\^ 


Monday 

Tuesday 

Wednesday 

Thursday 

Friday 

Saturday 

Salesmen's  totals 

Per  cents 


Smith 


I362.50 
415-75 
396.21 
472.96 
387.29 
493-89 


$2,528.60 


Brown 


J562.83 
475-92 
415-60 
516.29 
42936 
562.64 


32,962.64 


Jones 


I862.94 
732.83 
769-42 
640.20 
721.32 
816.25 


$4,542.96 


White 


5126.39 
143-62 
129.38 
145-17 
96.27 
163.92 


504.75 


Daily  Totals 


$1,914.66 
1,768.12 
1,710.61 
1,774-62 
1,634-24 
2,036.70 


$10,838.95 


Per  Cents 


17-6 
16.3 
15.8 
16.4 
15-1 


The  following  tabulation  illustrates  the  use  of  percentage  of 
increase  and  decrease  as  a  means  of  comparing  the  sales  of  each 
department  of  a  store  on  the  corresponding  days  of  two  years. 

Comparison  of  Sales  by  Departments 


Department 

Sales-Thursday 
December  is.  1919 

Sales-Thursday 
December  16,  1920 

Increase 
Decrease* 

% 
Increase — Decrease* 

A 

$      826.95 
1.034-78 
1.237.62 
2,643.80 
1,413.80 
962.40 
2,642.16 
1.964-39 
1,636.48 
1,213.42 

$      914-32 
1,231.64 
1,196.14 
2.843-27 
1.376.29 
1,235.96 
2,927.92 
2,129.80 
1,596.27 
1,723-96 

$      87-37 
196.86 

41.48* 
199.47 

37-51* 
273-56 
285-76 
165.41 

40.21* 
510.54 

10.57 
19.02 
3-35* 
7-54 
2.6s* 
28.42 
10.81 
8.42 
2.46* 
42.07 

B 

C 

D 

E 

f 

G 

H 

I 

J 

Total 

$15,575-80 

S17. 175-57 

Si, 599-77 

10.27 

The  average  of  a  number  of  quantities  may  be  accepted  as  the 
basis  of  comparison,  the  relation  of  each  quantity  to  the  average 


54  MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 

being  shown  in  terms  of  per  cents.  The  figures  in  the  preceding 
tabulation  of  "Monthly  Sales  Compared  on  a  Percentage  Basis" 
are  used  for  the  following  illustration: 

Individual  Sales  Compared  with  Average 

Name  Sales          Per  Cent  of  Average 

Arthur  Bradley $1,264.90  69.95 

J.B.Henderson   ....  1,913.52  105.82 

Fred  Bates 1,732.69  95.82 

Arthur  Button 2,213.72  122.42 

J.L.Weston 1,963.45  108.58 

Carter  Doane 1,627.32  89.99 

Walter' S.  Waite 1,692.18  93.58 

Frank  Chesley 2,138.45  118.26 

Harold  Peters 1,728.46  95-58 

Average $1,808.30  100.00 

Or  the  maximum  may  be  accepted  as  the  basis  of  comparison, 
the  relation  of  all  quantities  to  the  maximum  being  shown  in 
terms  of  per  cents.    Using  the  same  statistics  for  an  illustration: 

Individual  Sales  Compared  with  Maximum 

Name                            Sales  Per  Cent  of  Maximum 

Arthur  Bradley $1,264.90  57-i4 

J.B.Henderson 1,913.52  86.44 

Fred  Bates 1,732.69  78.27 

Arthur  Button 2,213.72  100.00 

J.  L.  Weston 1,963.45  88.69 

Carter  Boane 1,627.32  73-5i 

Walter  S.  Waite 1,692.18  76.44 

Frank  Chesley 2  138.45  96.60 

Harold  Peters 1,728.46  78.08 

The  following  tabulation  is  suggestive  of  the  use  which  may  be 
made  of  percentage  in  comparing  quantities  with  two  or  more 
similar  quantities  and  with  the  average  thereof.  In  this  case  the 
average  is  a  progressive  one. 


PERCENTAGE 

Comparison  of  Sales  of  Successive  Years 


55 


Year 

Sales 

Inc.  or  Dec* 

FROM  Preceding 

Year 

Inc.  or  Dec* 

from  First 

Year 

Progressive 

Average 
Prior  Years 

Inc.  or  Dec* 

FROM   Progressive 

Average 

Amount 

Amount 

rf 

/O 

Amount 

% 

1917 
1918 
1919 
1920 

$200,000 
238,000 
190,000 
285,000 

$38,000 
48,000* 
95.000 

19.00 

20.17* 
50.00 

$38,000 
10,000* 
85,000 

19.00 
s-oo* 
42.50 

$219,000 
209,333 

$29,000* 
75.667 

13.24* 
36.15 

Apportionment 

When  a  quantity  is  to  be  divided  or  partitioned,  the  basis  of 
the  partition  may  be  expressed  in  rates  per  cent.  The  partition 
is  then  accompHshed  by  applying  to  the  base  a  number  of  rates, 
the  total  of  which  is  100%.  The  division  of  partnership  profits 
is  a  familiar  illustration. 

Division  of  Profits 

Partners                              P.  &  L.  Ratio  Profits 

A 20%  $3,200.00 

B 35%  s,6oo.oo 

C 45%  7,200.00 

Total 100%  $16,000.00 

Or  the  apportionment  may  be  accomphshed  by  applying  the 
same  rate  to  a  number  of  bases  to  obtain  the  desired  percentages. 
In  this  case  the  rate  is  computed  by  dividing  the  total  percentage 
by  the  total  of  the  bases.  The  distribution  of  factory  overhead  is 
illustrative. 

Distribution  of  Factory  Overhead 

(Direct  Labor  Cost  Basis) 

Total  direct  labor,  all  departments $5,295.00  (base) 

Total  factory  overhead 3,460.00  (percentage) 

Then  3,460 -^  5,295  =  65.34+  %  (rate) 


56  MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 

Since  the  rate  is  approximate  only,  the  distribution  will  not 
be  exact;  a  remainder  of  24  cents  will  be  undistributed. 

Process  or  Proportion 

Department  Direct  Labor       of  Overhead 

I $2,140.00  $1,398.28 

2 1,965-00  1,283.93 

3 1,190.00  777-55 

Total $5.295-00  $3,459.76 

Gross  Profit  Method  of  Approximating  Inventory 

The  rate  of  gross  profit  of  prior  periods  may  be  used  to  ap- 
proximate an  inventory  when  it  is  impracticable  or  impossible  to 
take  a  physical  inventory.  This  is  accompHshed  by  utilizing  the 
elements  involved  in  the  computation  of  gross  profits.  In  a  bal- 
anced table,  when  all  but  one  element  is  known,  the  unknown 
element  is  found  as  the  amount  necessary  to  balance  the  table. 
The  gross  profit  on  sales  may  be  computed  by  setting  up  a  mer- 
chandise account  as  follows : 

Merchandise 

Inventory,  Jan.  i . .  .$100,000.00       Sales $350,000.00 

Purchases 300,000.00       Inventory,  Dec.  31 105,000.00 

The  gross  profit  would  be  $55,000.00,  the  amount  necessary  to 
bring  the  account  into  balance.  Now  if  the  inventory  were  not 
known,  but  the  gross  profit  could  be  estimated  at  $55,000.00, 
the  inventory  could  be  determined  thus: 

Merchandise 

Inventory,  Jan.  i $100,000.00    Sales $350,000.00 

Purchases 300,000.00 

Gross  profit 55,000.00 

The  inventory  would  be  $105,000,  the  amount  necessary  to 
bring  the  account  into  balance. 

Of  course  the  gross  profit  could  not  be  definitely  ascertained 
without  an  inventory,  but  it  could  be  approximated  by  using  the 


PERCENTAGE  57 

average  rate  of  gross  profit  on  sales  of  former  years,  if  no  radical 
variations  have  occurred  in  this  rate  and  if  there  is  no  reason  to 
believe  that  the  rate  of  the  current  period  has  been  radically 
different  from  the  average  rate  of  the  past. 

To  illustrate,  let  us  assume  that  the  sales  and  gross  profits  of 
the  business  whose  merchandise  account  appears  above,  were  as 
follows : 

Year  S.vles          Gross  Profit  Per  Cent 

Third  preceding  ....  $200,000.00         $31,400.00             15.7 

Second      "         ....  310,000.00           48,050.00             15.5 

First           "          ....  385,000.00            60,170.00             15.6 


,000.00       $139,620.00  15.6 


The  annual  rates  are  computed  to  determine  whether  there 
has  been  any  considerable  variation  in  the  rates  of  gross  profit. 
Then,  on  the  assumption  that  the  rate  of  gross  profit  for  the  cur- 
rent period  was  the  same  as  the  average  of  the  rates  of  the  three 
last  preceding  years: 

15.6%  of  $350,000.00  (sales)  =  $54,600.00,  approximate  gross  profit 

Then: 

Inventory,  January  i $100,000.00 

Add  purchases 300,000.00 

Total $400,000.00 

Deduct  cost  of  goods  sold  (approximate) : 

Sales $350,000.00 

Less  estimated  gross  profit 54,600.00       295,400.00 

Inventory,  Dec.  31  (approximate) $104,600.00 

The  inventory  thus  computed  is  $400  less  than  that  shown  by 
the  merchandise  account. 

The  following  problem  from  a  C.  P.  A.  examination  will  fur- 
ther illustrate  the  method,  the  principal  uses  of  which  are  in 


58  MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 

approximating  the  value  of  merchandise  destroyed  by  fire  and  in 
applying  the  gross  profit  test  to  the  verification  of  an  inventory. 

Problem 

The  accountant  is  called  on  to  confirm  the  inventory  of  a  mercantile 
establishment.  Investigation  shows  that  inventories  have  been  incor- 
rectly taken  and  are  padded.  It  is  mutually  agreed  that  all  inventories, 
except  the  first  one,  which  is  to  be  used  as  the  basis,  shall  be  entirely 
ignored. 

The  accountant  is  to  ascertain,  on  a  fixed  percentage  of  profit  which  it 
is  decided  shall  be  33yj%  of  sales,  what  stock  should  be  on  hand  December 
31,  1914,  with  the  following  data  obtained  from  the  various  books: 

Inventory  referred  to  as  a  basis,  January  i,  191 1 $47,350.29 

Gross  purchases  for  the  year  ending  December  31,  1911 .  .  .  76,320.15 

Returned  purchases 4,350.16 

Freight  and  drayage  on  purchases 325.14 

Gross  sales 115,469.31 

Returned  sales 1,317.12 

Gross  purchases  for  the  year  ending  December  31,  191 2  .  .  .  $65,506.80 

Returned  purchases 3,715.16 

Freight  and  drayage  on  purchases 41 7- 15 

Gross  sales 105,716.10 

Returned  sales 1,215.84 

Gross  purchases  for  the  year  ending  December  31,  1913  .  .  .  $62,517.10 

Returned  purchases 1,314.17 

Freight  and  drayage  on  purchases 316.17 

Gross  sales 101,317.18 

Returned  sales 1,216.06 

Gross  purchases  for  the  year  ending  December  31,  1914. .  .  .  $58,715.16 

Returned  purchases 287.50 

Freight  and  drayage  on  purchases 290.10 

Gross  sales 95,371.16 

Returned  sales 41 7- n 

Solution:  Since  the  rate  of  gross  profit  was  constant  throughout  the 
four  years,  and  since  only  the  final  inventory  is  required  by  the  problem,  the 
data  can  be  summarized  and  the  four  years'  totals  used  in  the  inventory 
calculation. 


PERCENTAGE 


59 


Summary  1911-1914 


Returned 

Returned 

Year 

Purchases 

Purchases 

Freight 

Sales 

Sales 

1911 

$76,320.15 

$4,350.16 

$325-14 

$115,469.31 

$1,317.12 

1912 

65,506.80 

3,715-16 

417-15 

105,716.10 

1,215.84 

1913 

62,517.10 

1,314-17 

316.17 

101,317.18 

1,216.06 

1914 

58,715-16 
$263,059.21 

287.50 

290.10 

95,371-16 

417. II 

Total 

$9,666.99 

$1,348.56 

$417,873.75 

$4,166.13 

Statement  of  Approximation  of  Inventory 
At  December  31,  1914 

Inventory,  January  i,  19 11 $47,350.29 

Add  cost  of  goods  purchased,  191 1-1914: 

Purchases $263,059.21 

Z,e55  returned  purchases  9,666.99  $253,392.22 


Add  freight. 


1,348.56     254,740.78  $302,091.07 


Deduct  cost  of  goods  sold,  1911-1914: 

Sales 

Less  returned  sales 


^17,873-75 
4,166.13  $413,707.62 


Less  gross  profit  (s^HVo  of  sales) 137,902.54     275,805.08 

Inventory,  December  31,  19 14  (estimated) $26,285.99 


Analysis  of  Statements 

The  following  statements  indicate  the  use  which  may  be  made 
of  percentage  in  analyzing  the  financial  statements  of  a  business 
to  show  such  facts  as  the  ratio  of  cost  of  sales,  expenses  and  profits 
to  sales;  the  relative  cost  of  the  various  elements  of  manufactured 
goods,  and  the  variation  in  operating  costs  of  different  years. 

Problem 

From  the  following  data  obtained  from  the  books  of  Johnson  and  Com- 
pany, construct  a  profit  and  loss  statement  showing  cost  of  goods  manufac- 
tured, and  cost  and  gross  profit  of  the  goods  sold.  Also  show  percentage  of 
each  eli^ment  based  upon  cost  of  manufacture  and  based  upon  sales. 


6o 


MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 


Raw  material,  January  i,  1915 $42,000,00 

"           "          December  31,  1915 45,000.00 

"           "          purchases  during  1915 130,000.00 

Freight  inward 4,218.00 

Wages  (productive) 70,000.00 

Sundry  manufacturing  expenses 3,500.00 

Sales 280,000.00 

Finished  goods,  January  i,  1915 18,000.00 

"               "       December  31,  191 5 22,000.00 

Selling  expenses 22,000.00 

Administrative  expenses 20,000.00 


Solution: 


Johnson  and  Company 

Profit  and  Loss  Statement 
Year  Ending  December  31,   1915 


Sales 1280,000.00 

Deduct: 

Cost  of  goods  sold: 
Raw  material: 

Inventory,  Jan.  i,  1915  .  .      $  42.000.00 
Purchases,  1915 130,000.00 

Total $172,000.00 

Inventory,  Dec.  31,  1915.  45,000.00     $127,000.00 

Freight  inward 4,218.00 

Productive  labor 70,000.00 

Manufacturing  expense.  .  ..  3,500.00 

Cost  of  goods  

manufactured $204,718.00 

Deduct: 

Inventory  variation — finished 
goods: 

December  31,  1915 $22,000.00 

January  i,  1915 i8,ooo.f)0  4,000.00 

Cost  of  goods  sold 200,718.00 

Gross  profit  on  sales 179,282.00 

Deduct  selling  expenses 22,000.00 

Net  profit  on  sales $57,282.00 

Deduct  administrative  expenses.  20,000.00 

Net  profit  on  operations $37,282.00 


%  OF      %  OF 

Cost   Sales 


62.0 

2.1 

34-2 

1.7 


71-7 

28.3 
7.9 


PERCENTAGE  6l 

This  illustration,  showing  the  per  cent  of  net  profit  and  gross 
profit  on  sales  raises  the  question  whether  sales  or  cost  of  sales 
should  be  used  as  the  base  in  the  computation  of  rates  of  gross 
profit.  In  common  parlance,  when  a  statement  is  made  that  a 
sale  has  resulted  in  realizing  a  certain  rate  of  profit,  the  rate  is 
understood  to  have  been  applied  to  the  cost.  Thus,  if  it  is  said 
that  an  article  costing  $2  was  sold  at  a  10%  profit,  one  assumes 
that  the  profit  was  20  cents  and  the  selling  price  $2.20.  But  in 
percentage  analyses  of  revenue  statements  it  is  much  more  con- 
venient to  use  the  net  sales  as  the  base.  Selling  expenses  nor- 
mally are  proportionate  to  sales  and  the  per  cent  of  selling  expense 
is  computed  on  the  basis  of  sales.  By  computing  the  cost  of  goods 
sold  and  afl  other  deductions  from  sales  as  percentages  of  the  net 
sales,  the  statement  begins  with  100%  as  the  base  and  continues 
on  the  same  basis  throughout.  But  if  cost  were  taken  as  the  base, 
the  sales  would  be  represented  by  a  rate  exceeding  100%,  and  the 
selling  expenses,  including  such  items  as  advertising,  salesmen's 
commissions,  freight  out  and  store  expense,  would  hsive  to  be 
rated  on  the  illogical  basis  of  cost. 

Therefore  the  percentage  analysis  of  the  revenue  statement 
should  properly  be  made  on  two  bases :  elements  of  manufacturing 
cost,  including  material,  labor  and  factory  overhead,  should  be 
considered  as  percentages  of  cost;  and  all  deductions  from  sales, 
including  the  cost  of  goods  sold,  the  selling  expenses  and  the 
administrative  expenses,  should  be  rated  as  percentages  of  the 
net  sales.    This  method  was  followed  in  the  preceding  illustration. 

As  an  illustration  of  the  use  of  percentage  in  the  comparison 
of  successive  revenue  statements,  the  following  condensed  state- 
ments of  the  J.  E.  Smith  Wire  and  Iron  Company  are  presented, 
together  with  comparative  percentage  analyses  thereof.  The 
analytical  statement  shows  that  the  increase  in  sales  and  the 
increase  in  cost  of  sales  have  not  been  proportionate,  as 
the  proportion  of  cost  to  sales  has  steadily  increased,  causing 
a  corresponding  decrease  in  the  percentage  of  profit. 


62 


MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 


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PERCENTAGE 


63 


As  another  illustration  of  the  utility  of  percentage  as  a  means 
of  measuring  variations,  the  following  condensed  statement  of 
comparative  manufacturing  costs  is  given. 

Condensed  Statement  of  Manufacturing  Costs 


Year  1916 

Year  1917 

D0LL.\RS 

%  of  Total 

Dollars 

%  of  Total 

$320,000.00 
441,500.00 
268,500.00 

31-07 
42.86 
26.07 

$287,600.00 
301,300.00 

187.100.00 

37.06 
38.83 

24.11 

Manufacturing  expense 

Total 

$1,030,000.00 

100.00 

$776,000.00 

While  this  statement  shows  the  variation  in  the  rate  per  cent 
of  each  item  to  the  total  cost,  it  does  not  give  an  adequate  idea 
of  the  variation  in  unit  costs.  The  following  table  shows  this 
variation. 

Comparative  Statement  of  Unit  Costs 


Units  Produced: 
1916 
1917 


10.000 
8,000 


Year  1916 

Year  1917 

In'crease  or 
Decrease* 

Total 

Unit 

Total 

Unit 

Dollars 

Per  cent 

$320,000.00 
441,500.00 
268,500.00 

$32.00 
44-15 
26.85 

$287,600.00 
301,300.00 
187,100.00 

$35-95 
37-66 
23-39 

$3  95 
6.49* 
3-46* 

12.34% 

14.70* 

12.89* 

Manufacturing  expense 

Total 

$1,030,000.00 

S103.00 

$776,000.00 

$97-00 

$6.00* 

S-83* 

Percentage  Analyses  to  Determine  Causes  of  Variation  in  Profits 

Revenue  statements  may  be  compared  on  a  percentage  basis 
to  show  the  cause  of  the  increase  or  decrease  in  net  profit,  but  in 
order  to  arrive  at  accurate  results  it  is  necessary  to  know  the  per 


64 


MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 


cent  of  increase  or  decrease  in  the  unit  selling  price  of  the  second 
year  as  compared  with  the  unit  selling  price  of  the  first  year.  To 
illustrate,  let  us  assume  the  following  statements  each  of  which 
shows  a  percentage  analysis  based  on  sales: 

The  Wharton  Manufacturing  Company 

Comparative  Profit  and  Loss  Statements 

Year  Ending  Year  Ending 

December  31,  1916  December  31,  1917 

Sales $400,000  100.00%  |6oo,ooo  100.00% 

Deduct: 
Cost  of  sales: 

Material Iioo.ooo  25.0  $150,000  25.0 

Labor 200.000  50.0  330,000  55.0 

Manufacturing 

expense 50,000  12.5  60,000  10. o 

Total 350,000  87.50  540,000  00.0 

Gross  profit  on  sales ...  .  $50,000  12.50  $60,000  10. o 

DedMci  selling  expense .  io,ooo  2.50  IS, 000  2.5 

Net  profit  on  sales $40,000  10.00  $45,000  7.5 

/JeiiMc/ general  expenses  15,000  3.75  15,000  2.5 

Net  profit $25,000  6.25  $30,000  5.0 

This  comparative  statement  has  very  little  meaning  until  the 
further  fact  is  known  that  there  was  a  20%  advance  in  the  selling 
price  of  all  goods  in  191 7.  With  this  additional  information,  the 
sales  of  191 7  can  be  reduced  to  the  value  which  they  would  have 
brought  in  19 16,  thus: 

$600, ooo-^  1 20%=  $500,000,  the  price  which  the  same  goods  would  have 

sold  for  in  iqi6 

A  supposititious  profit  and  loss  statement  for  191 7  may  now  be 
drawn  up,  beginning  with  sales  of  $500,000  (volume  of  191 7  at 
19 1 6  prices)  and  replacing  the  actual  figures  of  191 7  with  amounts 
obtained  by  multiplying  $500,000  by  the  various  rates  per  cent 
shown  in  the  1916  statement.  This  supposititious  statement  will 
show  what  the  figures  for  191 7  would  have  been  if  there  had  been 
merely  a  change  in  volume  of  business  but  no  change  in  the  rates 
of  the  various  expenses  to  sales.  A  comparison  of  these  sup- 
posititious figures  with  the  actual  figures  for  191 7  will  show  the 


PERCENTAGE 


65 


effect  on  191 7  profits  of  fluctuations  in  cost  of  manufacture  and 
of  expenses. 

Profit  and   Loss  Statement  of   191 7   Reduced  to   1916  Rates  and 
Compared  with  Actual  Statement  for  191 7 

Statement  on  Basis 

OF  1916  Rates  to  Sales  ,,  _ 

Variations  of  Profit 
Amounts 
Rates  of     at  1916        Actual  Decreases  Increases 

1916  Rates  1917  Dollars      %  Dollars     % 

Sales 100.00%  Jsoo, 000  S6oo,ooo                                    Jioo.ooo     20.0 

Deduct: 

Cost  of  goods  sold: 

Material 23.00        125,000  150,000       $25,000     20.0 

Labor 50.00       250,000  330,000          80,000     32.0 

Manufacturing  expense.  .  .  12.50         62,500  60,000                                             2,500       4.0 

Total 87.50     $437,500     J540.000     $102,5002343 

Gross  profit 12.50       862,500       56o,ooo         $2,500     4.00 

Z?e<iuc<  selling  e.xpenses ..  .  2.50  12,500  15,000  2.500  20.00 

Net  profit  on  sales 10.00       $50,000       $45,000  $5,000   10.00 

Deduct  geneval  expenses . .  3.75  18,750  15,000  3,750     200 

Net  profit 6.25        $31250       $30,000  $1,250     4.00 


The  additional  profit  of  191 7  can  now  be  accounted  for  as 
follows : 

i2>^%  gross  profit  (1916  rate)  on  $100,000.00  ad- 
ditional business  done  in  191 7 $1  2,500.00 

Deduct  excess  of  additional  cost  to  manufacture 
over  increase  in  selling  prices: 

Additional  cost  of  material $25,000.00 

Additional  cost  of  labor 80,000.00 

Total $105,000.00 

Less  decrease  in  cost  of    manufacturing 

expense 2,500.00 

Net  excess  in  manufacturing  cost $102,500.00 

Less  increase  in  selling  prices 100,000.00         2,500.00 

Excess  of  gross  profit  of  191 7  over  191 6 $10,000.00 

Deduct  addhional  selling  expenses  of  191 7.  .  .  5,000.00 

Additional  net  profit  of  191 7 $5,000.00 


CHAPTER   VII 

EQUATIONS  IN  THE  SOLUTION  OF  PROBLEMS 

Solving  Equations 

Problems  may  often  be  solved  by  stating  the  conditions  in  the 
form  of  an  equation  and  solving  the  equation  by  applying  one  or 
more  of  the  following  processes : 

1.  Multiplying  both  sides  of  the  equation  by  the  same 

number 

2.  Dividing  both  sides  of  the  equation  by  the  same  number 

3.  Adding  the  same  number  to  both  sides  of  the  equation 

4.  Subtracting  the  same  number  from  both  sides  of  the  equa- 

tion 

Illustration  i 

A  manufacturer  produced  a  certain  commodity  which  he  sold  the  first 
year  at  a  certain  price;  he  raised  the  price  25%  the  second  year;  increased 
that  price  20%  the  third  year;  and  in  the  fourth  year  he  increased  the 
third-year  price  by  1673%.  The  price  the  fourth  year  was  $35.  What 
was  the  price  the  first  year? 

Solution:  In  order  to  obtain  an  equation  it  is  necessary  to  represent 
some  value  by  100%.  The  value  chosen  to  be  represented  by  100%  will 
depend  on  the  conditions  of  the  problem;  where  convenient,  100%  should 
represent  the  value  required  by  the  problem.     In  this  case — 

Let  100%  =  the  selling  price  the  first  year 

then  100%  X  125%     =  125%  the  selling  price  the  second  year 
and  125%  X  120%       =150%    "       "  "       "   third        " 

and  150%  X  ii6V3%  =  i75%    "       "  "       "   fourth     " 

Since  the  selling  price  the  fourth  year  was  $35  we  obtain  the  equation: 

175%  =  l3S 
66 


EQUATIONS  IN  THE  SOLUTION  OF  PROBLEMS     67 

But  it  is  desired  to  determine  100%,  which  is  accomplished  by  dividing 
both  ?ides  of  the  equation  by  1.75: 

100%  =  $20,  the  selling  price  the  first  year 

Illustration  2 

A  entered  into  partnership  with  B  and  was  to  act  as  manager  of  the 
business.  Before  dividing  profits  equally  with  B,  A  was  to  receive  a 
special  bonus  of  25%  of  the  net  profit.  Before  calculating  A's  commission, 
the  profits  were  shown  by  the  revenue  statement  to  be  $5,000.  How 
should  the  $5,000  be  divided  between  A  and  B? 

Solution:  This  problem  illustrates  the  difficulty  which  frequently 
arises  in  interpreting  contracts  which  provide  that  commissions  and 
bonuses  shall  be  determined  as  percentages  of  the  net  profit.  The  difii- 
culty  arises  from  the  uncertainty  as  to  what  is  the  amount  of  the  net 
profit;  for,  if  the  bonus  is  to  be  considered  as  an  expense  of  the  business, 
the  net  profits  are  less  than  $5,000;  if  the  bonus  is  not  to  be  considered 
as  an  expense  but  as  part  of  the  distribution  of  profits,  the  net  profit 
is  $5,000.  From  the  statement  of  the  problem,  it  is  impossible  to  tell 
whether  or  not  the  commission  is  to  be  considered  as  an  expense;  hence  it 
is  necessary  to  give  two  solutions. 

Assuming  the  bonus  is  not  an  expense: 

Since  the  bonus  is  25%  of  the  net  profit,  the  net  profit  must  be  100%. 
Since  the  bonus  is  not  an  expense,  the  $5,000  is  all  to  be  considered  net 
profit — 

Then,  100%  =  $5,000,  the  net  profit 
and        25%  =     1,250,  the  bonus 

and        75%  =  $3,750,  the  remaining  profit  to  be  divided  equally. 
37^2%  =  $1,875,  share  to  A  and  B  each 

Therefore,  the  division  is  as  follows: 

A  B  Total        Rate 

Bonus $1,250  $1,250         25% 

Remainder,  K  each 1,875     $1,875        3,75°  75% 


Total $3,125     $1,875      $5,000        100% 


68  MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 

Assuming  that  the  bonus  is  an  expense  to  be  deducted  from  the  $5,000 
to  obtain  the  net  profit: 

25%  =  bonus 
100%  =  net  profit 


125%  =  bonus  plus  profit,  or  $5,000 

Then  100%  =  $4,000,  net  profit 
and        25%  =  $1,000,  bonus 


Proof  125%  =  $5,000 


The  division  of  the  $5,000  on  this  assumption  would  be: 

A                B          Total  Rate 

Gross  profit $5,000  125% 

Deduct  bonus $1,000                      1,000  25% 


Net  profit,  y2  each 2,000    $2,000     $4,000        100% 

Total  distribution $3,000    $2,000 


The  net  difference  of  $125  in  the  distribution  under  the  two 
interpretations  emphasizes  the  necessity  for  care  in  drawing  up 
contracts  of  this  character.  In  such  cases  an  accountant  should 
always  be  consulted  as  to  the  wording  of  the  contract,  as  any 
competent  accountant  would  recognize  the  danger  of  there  being 
two  constructions  placed  on  the  contract,  with  a  consequent 
dispute. 

Illustration  3 
(From  Ohio  C.  P.  A.  Examination,  October,  1919) 

The  American  Manufacturing  Company  commenced  business  on 
January  i,  1918,  with  a  paid-up  cash  capital  equal  to  the  sales  for  the  year 
1918. 

The  net  profits  for  the  year  1918  were  $26,100. 

Of  the  total  charges  to  manufacturing  during  the  year,  40%  was  for 
materials,  30%  for  productive  labor,  and  30%  for  manufacturing  ex- 


EQUATIONS  IN  THE  SOLUTION  OF  PROBLEMS     69 

penses  (including  5%  depreciation  on  plant  and  machinery,  amounting  to 
$3,000). 

The  value  of  the  materials  used  was  80%  ot  the  amount  purchased,  and 
90%  of  the  amount  purchased  was  paid  during  the  year. 

The  inventory  value  of  finished  goods  on  hand  at  December  31,  1918, 
was  10%  of  the  cost  of  finished  units  delivered  to  the  warehouse,  and  the 
work  in  process  at  that  date  was  equal  to  50%  of  the  cost  of  units  delivered 
to  the  warehouse. 

The  selling  and  administrative  expenses  were  equal  to  20%  of  the 
sales;  also  to  40%  of  the  cost  of  goods  sold.  Ninety  per  cent  of  these  ex- 
penses were  paid  during  the  year  1918.  Plant  and  machinery  purchased 
during  the  year  were  paid  for  in  cash. 

All  labor  and  manufacturing  expenses  (exclusive  of  depreciation)  were 
paid  in  full  up  to  and  including  December  31,  1918. 

Of  the  total  sales  for  the  year,  80%  was  collected  and  1%  charged 
ofT  as  worthless. 

From  the  given  data  you  are  required  to  prepare  a  balance  sheet  and  a 
profit  and  loss  statement,  showing  cost  of  goods  delivered  to  the  warehouse, 
cost  of  goods  sold,  and  net  profit  for  the  year. 

Solution: 
Let  sales  equal  100% 

Then  (since  selling  and  administrative  expense  is 
20%  of  sales  or  40%  of  cost  of  goods  sold)  the 
cost  of  goods  sold  is  half  of  the  sales  or  50% 

and  the  gross  profit  is  50% 

(That  is,  if  a  is  20%  of  x  and  is  also  40%  of  y,  then  y 

must  be  half  of  x.) 
Selling  and  administrative  expenses  are  equal  to       20% 

and  bad  debts  equal  1%       21% 

Hence  the  net  profit  is  29% 

Then,     29%  =  $26,100 

and      100%  =  $90,000,  sales  for  the  year,  and  the  cash  capital  at 
the  beginning  of  the  year. 

Cost  of  goods  sold  =  50%  of  $90,000 $45,000 

Gross  profit 45,000 

Selling  and  administrative  expense 18,000 

Bad  debts 900 


70  MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 

Since  the  inventory  of  finished  goods  at  December  31,  i9i8,wasio%of 
the  cost  of  finished  units  delivered  to  the  warehouse,  the  cost  of  goods  sold 
was  90%  of  the  finished  goods  manufactured  during  the  year. 

Then,  $45,000-7-90%  =  $50,000,  cost  of  finished  goods  manufactured 
And     $50,000  —  $45,000  =  $5,000,  inventory  of  finished  goods  at  Decem- 
ber 31,  1918 

Since  the  work  in  process  at  December  31,  191 8,  was  50%  of  the  cost 
of  finished  goods  delivered  to  the  warehouse, 

50%  of  $50,000  =  $25,000,  Work  in  process  inventory 

$50,000,  Goods  finished  during  1918 
25,000,  Work  in  process  at  Dec.  31,  1918 

$75,000,  Total  manufacturing  cost  of  1918 

40%  of  $75,000.00  =  $30,000,  Cost  of  materials  used 
30%  of    75,000.00  =     22,500     "     "   productive  labor 
30%  of    75,000.00  =     22,500     "     "   manufacturing  expense 

Of  this  manufacturing  expense,  $3,000  was  depreciation  on  plant  and 
machinery;  hence  the  manufacturing  expense  paid  in  cash  was  $19,500. 
Since  the  rate  of  depreciation  was  5%,  the  cost  of  plant  and  machinery  was 
$60,000,  all  of  which  was  paid  for  in  cash. 

Since  80%  of  the  material  purchased  was  used  in  manufacturing, 
$30,000-^-  80%  =  $37,500,  the  cost  of  the  material  purchased;  and  $7,500 
is  the  inventory  of  raw  material  at  December  31,  191 8.  Also  90%  of 
$37,500,  or  $33,750,  is  the  amount  of  cash  paid  for  purchases;  and  $37,500 
~~  ^33-75°  =  $3,750-  the  accounts  payable  at  December  31,  1918,  for 
purchases. 

The  selling  and  administrative  expenses  were  $1 8,000.  Of  this  amount, 
90%,  or  $16,200,  was  paid  in  cash.  The  remainder,  $1,800,  is  an  addition 
to  the  accounts  payable. 

80%  of  the  sales  of  $90,000  were  collected.  1%  was  written  off. 
Hence, 

Sales $90,000 

Less: 

Cash  collections $72,000 

Bad  debts 900       72,900 

Balance  of  accounts  receivable.  .  .  .  $17,100 


EQUATIONS  IN  THE  SOLUTION  OF  PROBLEMS  71 

The  cash  summary  is : 

Cash  capital  paid  in $90,000 

Collections  on  accounts  receivable 72,000    $162,000 

Deduct: 

Plant  and  machinery $60,000 

Materials 33-75° 

Productive  labor 22,500 

Manufacturing  expense IQ.500 

Selling  and  administrative  expense 16.200 

Total  disbursements 151,950 

Balance $  10,050 


American  Manufacturing  Company 

Trial  Balance 
December  31,  191S 

Capital  stock $90,000 

Plant  and  machinery $60,000 

Reserve  for  depreciation,  plant  and  machinery ....  3,000 

Sales 90,000 

Purchases 3  7,500 

Productive  labor 22,500 

Manufacturing  expense 19,500 

Depreciation,  plant  and  machinery 3,000 

Selling  and  administrative  expense 18,000 

Bad  debts 900 

Accounts  receivable 17,100 

Accounts  payable  (3,750  +  1,800) 5, 550 

Cash 10,050 

$188,550    $188,550 

Inventories:  raw  material,  $7,500;  goods  in  process,  $25,000; 
finished  goods,  $5,000. 


72 


MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 


American  Manufacturing  Company 

Profit  and  Loss  Statement 
Year  Ending  December  31,  1918 

Sales $90,000.00 

Deduct: 

Cost  of  goods  sold : 
Material: 

Purchases $37,500.00 

Less  inventory,  Dec.  31, 

igi8 7,500.00     $30,000.00 

Productive  labor 22,500.00 

Manufacturing  expense 19,500.00 

Depreciation — plant        and 

machinery 3,000.00 

Total  manufacturing  cost .  $75,000.00 

Deduct   goods   in    process — 

Dec.  31,  1918 25,000.00 

Cost  of  finished  goods  manu- 
factured   $50,000.00 

Deduct    inventory     finished 

goods — Dec.  31,  1918 5.000.00       45,000.00 

Gross  profit  on  sales $45,000.00 

Deduct: 

Selling  and  administrative  ex-  $18,000.00 

pense 

Bad  debts 900.00       18,900.00 

Net  profit $26,100.00 


EQUATIONS  IN  THE  SOLUTION  OF  PROBLEMS 


73 


American  Manufacturing  Company 


Assets 

Plant  and  machinery  . . 
Less  depreciation  .  .  . . 

Balance  Shej 
December  31,   i 

.       160,000.00 

3,000.00     $  57,000.00 

ET 
918 

Liabilities 

Accounts  payable  .  .  . 

Raw  material 

Goods  in  process 

7,500.00 
25,000.00 

5.000.00 
17,100.00 
10,050.00 

5,550.00 

Accounts  receivable  . .  .  . 
Cash 

$121,650.00 

$121,650.00 

Illustration  4 

As  another  illustration  of  the  use  of  percentage  in  the  solution  of  prob- 
lems, the  following  C.  P.  A.  problem  is  given. 

The  Orinoco  Coal  Company  was  incorporated  under  the  laws  of  the 
state  of  Illinois,  with  an  authorized  capital  of  $8,000,  divided  into  eighty 
shares  of  the  par  value  of  f  100  each,  which  were  subscribed  for  as  follows: 

Samuel  Black 60  shares 

William  Green 10      " 

John  White      10      " 

Samuel  Black  was  elected  president;  George  Brown,  vice-president  and 
manager,  and  Charles  Pinck,  secretary  and  treasurer.  Neither  Brown  nor 
Pinck  held  any  stock,  but  were  to  receive  in  addition  to  their  salaries  a 
percentage  of  the  profits  after  charging  off  all  losses  from  whatever  source 
— Brown  15%  and  Pinck  10%.  These  shares  in  the  profits  were  not  to 
be  considered  an  expense  deductible  to  obtain  the  basis  of  the  bonuses. 

At  the  end  of  the  year  a  meeting  of  the  stockholders  was  held  and 
the  following  balance  sheet  was  presented: 


Assets 

Liabilities 

Cash 

$  8,031.12 

Accounts  payable .... 

$       740.22 

Accounts  receivable . . 

817-32 

Reserve  for  bad  debts 

376.05 

Coal 

6,644-15 

Capital  stock 

8,000.00 

Charles  Pinck 

2,264.14 

Undivided  profits .... 

8,640.46 

$17,756.73 

$17,756.73 

74  MATHEMATICS  OP  ACCOUNTING  AND  FINANCE 

It  was  announced  that  Treasurer  Pinck,  who  was  not  financially 
responsible  and  was  not  bonded,  had  disappeared  and  his  account  was 
uncollectible,  but  would  be  reduced  by  crediting  the  account  with  his  share 
of  the  net  profits. 

What  amounts  should  Pinck  and  Brown  receive  as  salary  addition? 

What  is  the  amount  of  Pinck's  defalcation? 

If  the  remaining  profit  is  divided  among  the  stockholders,  what  divi- 
dend should  each  receive? 

Solution:  Since  the  bonuses  allowable  to  Pinck  and  Brown  are  to  be 
calculated  on  the  net  profit  after  deducting  all  losses,  the  defalcation  must 
be  deducted  from  the  undivided  profits  of  $8,640.46  to  obtain  the  net 
profit. 

Let  100%  =  the  net  profit, 

then  $8,640.46  —  defalcation  =   100%. 

But  the  amount  of  the  defalcation  is  not  known,  except  that  it  is  the 
amount  of  Pinck's  debit  balance,  $2,264.14,  minus  his  10%  of  the  net 
profit ;  hence — 

$2,264.14  —  10%  =  defalcation 

Substituting  the  first  term  of  this  equation  for  "defalcation"  in  the  first 
equation,  we  obtain — 

$8,640.46  —  ($2,264.14  —  10%)  =  100% 

Removing  the  parentheses  and  changing  signs — 

$8,640.46  —  $2,264.14+  10%  =  100% 

Subtracting  10%  from  both  sides  of  the  equation — 

$8,640.46  —  $2,264.14  =  90% 
or  $6,376.32  =  90% 

Dividing  both  sides  by  90% — 

$7,084.80  =  100%,  the  net  profit 
Then  708.48  =  10%,  Pinck's  bonus 
and       1,062.72  =     15%,  Brown's  bonus 

$2,264.14,  Pinck's  debit  balance 
708.48      "  bonus  credited 


'1)555-66      "  defalcation 


EQUATIONS  IN  THE  SOLUTION  OF  PROBLEMS      75 

$8,640.46  undivided  profit  per  balance  sheet 
1,555.66  Pinck's  defalcation 


$7,084.80  net  profit — basis  of  Brown's  and  Pinck's  bonus 
$708.48  Pinck's  bonus 

1,062.72       1,771.20  Brown's  bonus  and  total 

$5,313.60  profits  remaining  for  dividends 


Vs  of  $5,313.60  =  $3,985.20,  Black's  dividend 
yg "         "  =      664.20,  Green's      " 

1/8 "         "  =      664.20,  White's      " 

i/g"         "  =  $5,313.60,  total  (as  above) 


CHAPTER  VIII 
TRADE  AND  CASH  DISCOUNT 

Trade  Discount 

There  are  two  kinds  of  discount  affecting  the  amount  received 
for  goods  sold  or  the  amount  paid  for  goods  bought.  The  first 
of  these  is  trade  discount,  which  is  a  device  for  varying  prices 
without  interfering  with  basic  or  *'hst"  prices,  often  called  retail 
prices.  The  convenience  of  its  use  arises  from  the  fact  that  an 
expensive  catalogue  can  be  made  permanent  by  recording  only  the 
list  prices.  The  real  or  trade  prices  are  determined  by  the  trade 
discounts  offered  by  the  seller,  which  are  usually  contained  in 
confidential  letters  or  circulars  sent  to  customers. 

A  great  saving  of  expense  is  effected  by  not  having  to  issue  a 
new  catalogue  whenever  market  prices  are  modified.  Moreover 
the  printing  and  circulating  of  large  catalogues  would  take  con- 
siderable time  so  that  it  would  be  impossible  to  give  effect  to 
price  revisions  until  weeks  after  the  necessity  for  them  had  arisen. 
But  a  circular  altering  the  trade  discount  and  thus  raising  or 
lowering  the  actual  prices,  can  be  prepared  on  a  reproducing  ma- 
chine and  sent  out  to  customers  in  one  or  two  days. 

The  way  in  which  a  trade  discount  operates  is  as  follows:  A 
manufacturer  or  wholesaler  sells  to  a  retailer  at  75  cents  an  article 
the  list  or  gross  price  of  which  is  $1.25.  He  bills  the  article  to  the 
buyer  at  the  gross  price  but  then  deducts  the  discount  of  40%, 
thus: 

16  dozen  of  article      I15.00     $240.00 

Less  40%  96.00     $144.00 


In  this  way  the  actual  or  net  price  of  the  article  isestabhshed 
at  75  cents.    This  actual  price  is  the  amount  entered  in  the  books 

76 


TRADE  AND  CASH  DISCOUNT  77 

of  both  seller  and  purchaser,  neither  of  whom  makes  any  record 
of  the  list  price  or  the  trade  discount. 

Cumulative  Trade  Discounts 

In  order  to  provide  for  fluctuations  in  price  the  device  of 
cumulative  trade  discounts  has  been  adopted.  Thus  the  dis- 
counts quoted  may  be  30,  20,  10,  and  5.  This  does  not  mean  a 
total  discount  of  65,  because  each  successive  rate  is  calculated 
on  the  amount  left  after  deducting  the  discount  at  the  preceding 
rate.  Thus,  if  the  list  price  is  $240,  and  the  discount  is  30,  20, 
10,  and  5,  the  computation  is  as  follows: 

List  price $240.00 

Less  30%  of  $240.00 72.00       $72.00 

$i6S.oo 
Less  20%  of  $168.00 3360        33.60 

$134.40 
Less  10%  of  $134.40 13.44         13-44 

$120.96 
Less    5%  of  $1 20.96 6.05  6.05 

$114-91     $125.09 

The  total  discount  of  $125.09  is  almost  exactly  52  3^  %  of  the 
gross  price  of  $240.    The  net  or  real  price  is  $1 14.91 . 

If  the  wholesaler  wishes  to  advance  the  price,  he  notifies 
the  trade  that  the  last  discount  of  5%  is  discontinued,  which 
will  make  the  amount  $120.96,  or  that  the  last  two  discounts 
of  10  and  5  are  replaced  by  5  alone,  raising  the  price  to  $127.68 
($134.40  -  $6.72). 

If  the  price  is  to  be  lowered,  it  is  done  by  adding  a  further  dis- 
count. If  another  5  is  added  the  net  price  becomes  $109.16 
($114.91  -  $5.75). 

Methods  of  Finding  Net  Price 

In  order  to  avoid  the  necessity  of  making  a  separate  computa- 
tion for  each  discount,  it  is  well  to  know  how  to  find  one  rate  that 


78  MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 

will  give  the  same  result  as  that  reached  by  the  successive  steps  of 
the  combination  rate.    The  rule  for  this  is : 

Add  the  first  two  discounts;  multiply  the  two  discounts;  sub- 
tract the  second  result  from  the  first.  With  this  result  as  one  dis- 
count combine  the  third  in  the  same  manner;  and  take  up  in  turn 
each  of  the  other  discounts. 

Thus  in  the  illustration — 

The  sum  of  .30  and  .20  is  .50 

The  product  of  .30  and  .20  is  .06  .44 


The  sum  of  .44  and  .10  is  .54 

The  product  of  .44  and  .10  is  .044  .496 


The  sum  of  .496  and  .05  is  .546 

The  product  of  .496  and  .05  is  .0248         .5212 


Therefore  the  combination  rate  is     .5212 


It  is  not  necessary  to  remember  any  rule,  because  this  same 
result  can  be  reached  by  computing  the  discount  on  $100  thus: 

List  price $100.00 

Less  30% 30.00     .30 

$70.00 
Less  20% 14.00     .14 

$56.00 
Less  10% 5.60     .056 

$50.40 
Less    s% 2-52     .0252 

Net  price  and  total  discount  rate. .     $47.88     .5212 

This  latter  method  has  the  further  advantage  of  being  sus- 
ceptible of  easy  proof,  since  the  net  price  and  the  discount  must 
add  to  100. 

A  still  better  way  of  reaching  the  same  result  is  to  compute  the 
net  amount  at  once,  instead  of  finding  the  discount  and  deduct- 


TRADE  AND  CASH  DISCOUNT  79 

ing  it.  Thus,  instead  of  ascertaining  that  30%  of  $240  is 
$72  and  deducting  it  to  find  $168,  it  is  shorter  to  multiply  240 
by  70%,  or  rather  by  .7,  to  get  the  same  result,  and  then  to 
multiply  168  by  .8  to  get  134.40,  and  so  on.  In  other  words,  mul- 
tiply each  successive  amount  by  the  complement  of  its  discount 
rate,  and  the  final  result  will  be  the  net  price,  with  all  discounts 
deducted. 

In  order  to  find  a  single  rate  for  the  net  amount,  multiply  all 
the  complements  of  the  discount  rates.  Thus,  the  product  of  .7  X 
.8  X  .9  X  .95  is  .4788,  the  single  rate  for  the  net  price  when  the 
discounts  are  30,  20,  10,  and  5.  Applying  this  single  rate  to  $240 
will  give  $114.91,  the  same  result  as  by  the  other  method. 

It  will  save  a  great  deal  of  labor,  as  well  as  insure  greater  ac- 
curacy, if  tables  are  made  of  the  net  amounts  resulting  from  the 
application  of  the  discounts  usually  given  by  the  concern.  These 
tables  should  be  made  for  amounts  of  $1  to  $100  and  should  be 
built  up  by  successive  additions  and  not  multiplications.  They 
will  prove  at  every  tenth  amount,  since  if  the  net  for  $1  is 
$.4788,  it  will  be  $4.7880  for  $10,  thus  proving  every  intermediate 
amount,  whereas  there  is  no  proof  if  each  amount  is  found  by 
independent  multiplication. 

It  is  not  necessary  to  carry  the  tables  beyond  $100  if  all  the 
decimals  are  used.  It  is  necessary  only  to  be  careful  to  move  the 
decimal  point  enough  spaces  to  the  right  to  represent  the  higher 
numbers.  Thus,  if  the  list  price  is  $4,836  and  the  net  rate  .4788, 
the  table  will  show  on  line  36  the  amount  of  17.2368  and  on  line 
48  the  amount  of  22.9824.    Therefore: 

The  net  of  $      36.00  is  $      17.2368 
"     "     "       4,800.00  "     2,298.24 


"     "     "     $4,836.00  "  $2,315.48 

The  same  result  could  be  reached  with  a  table  of  only  ten  lines, 
from  $1  to  $10,  but  a  table  of  100  lines  shortens  the  computa- 
tions and  is  still  easily  contained  on  a  comparatively  small  card. 


8o  MATHEMATICS  OP  ACCOUNTING  AND  FINANCE 

Cash  Discount 

In  order  to  induce  prompt  payment  of  accounts  merchants 
frequently  offer  to  deduct  a  certain  per  cent  from  bills  if  they  are 
paid  within  a  fixed  number  of  days.  This  deduction  is  called  a 
cash  discount  and  is  always  applied  to  the  net  or  trade  price 
reached  after  all  trade  discounts  have  been  deducted. 

The  discount  terms  are  expressed  by  the  figure  of  the  per  cent 
followed  by  the  number  of  days  in  which  the  discount  is  allowed 
and  then  by  the  total  number  of  days  that  may  elapse  before  the 
bill  becomes  due,  thus: 

2/ 10 ;  1/30 ;  N/60 

which  means  that  if  payment  is  made  in  10  days  2%  may  be  de- 
ducted; if  paid  in  30  days  1%;  and  that  the  bill  is  due  in  60  days 
net,  that  is,  without  any  discount. 

The  advantages  of  giving  cash  discount  are  usually  said  to  be 
that  they  decrease : 

1.  Loss  from  bad  debts 

2.  Cost  of  collecting  accounts 

3.  Amount  of  capital  tied  up  in  outstanding  accounts 

As  only  those  who  are  comparatively  strong  financially  are 
able  to  avail  themselves  of  discounts  offered,  the  first  advantage 
would  not  appear  to  be  very  often  realized. 

The  advantage  to  the  purchaser  is  that  he  makes  much  more 
than  normal  interest  by  taking  his  discount.  Thus,  in  the  case  of 
2/10;  1/30;  N/60,  if  the  2%  is  taken  the  purchaser  gains  2%  for 
50  days'  use  of  the  money,  which  is  at  the  rate  of  14.6  per  cent  per 
annum,  while  if  only  1%  is  taken  he  gains  1%  for  30  days,  or  1 2% 
per  annum.  If  he  has  sufficient  bank  credit,  he  can  well  afford 
to  borrow  at  6  or  7%  in  order  to  take  his  discounts. 

Discount  as  a  Protection  against  Loss 

The  term  "cash  discount"  is  usually  understood  to  refer  to  de- 
ductions that  amount  to  a  rather  heavy  interest.    In  some  cases, 


TRADE  AND  CASH  DISCOUNT  8l 

however,  the  deduction  allowed  is  far  more  than  this.  For  in- 
stance, one  concern  manufacturing  electrical  apparatus  sells  on 
terms  of  40%  discount  if  paid  within  30  days.  If  not  paid  within 
the  time  the  price  is  at  list.  So  large  a  discount  is  not  usually 
considered  to  come  within  the  definition  of  a  cash  discount,  al- 
though it  is  such,  strictly  speaking,  since  it  is  dependent  upon 
the  payment  of  cash.  It  would  perhaps  be  better  to  call  it  a 
trade  discount  with  a  time  limit. 

A  discount  of  this  kind  is  adopted  as  a  protection  against  loss 
in  case  of  the  bankruptcy  of  a  customer.  If  a  discount  is  given 
purely  as  a  trade  discount,  a  price  is  established  that  remains  the 
same  whether  a  bill  is  paid  at  maturity  or  not.  If  the  customer 
becomes  bankrupt,  the  claim  filed  with  the  receiver  must  be  the 
net  with  all  trade  discounts  deducted.  If  the  list  price  is  $1,000 
and  the  unconditional  trade  discount  is  40%,  the  claim  must 
be  filed  for  $600.  If  the  final  settlement  is  for  50%,  the  creditor 
loses  $300.  But  if  the  discount  has  a  time  limit  of  30  or  60  days, 
the  time  will  have  expired  and  a  claim  can  be  filed  for  the  list 
price  of  $1 ,000,  on  which  the  dividend  will  be  $500.  The  creditor 
will  lose  only  $100,  instead  of  $300. 

Cash  Discount  Regarded  as  an  Expense 

A  view  of  cash  discount  not  very  generally  accepted  is  that 
the  net  price  is  the  real  price  and  that  if  the  bill  is  not  paid  in 
time  the  discount  is  added  as  a  penalty.  This,  of  course,  reverses 
the  usual  understanding  of  the  subject,  which  is  that  discount 
taken  is  a  profit.  If  this  view  is  adopted,  cash  discount  taken 
is  eliminated  entirely,  and  discount  not  taken  becomes  an  ex- 
pense. 

To  illustrate,  if  the  trade  price  is  $1,000  with  an  option  of  a 
cash  discount  of  2%,  the  entries  would  be  as  follows: 

Purchases $980 

Cash  discount 20 

Creditor $1,000 


82  MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 

Then  if  discount  is  taken, 

Creditor $i,ooo 

Cash $980 

Cash  discount 20 

In  this  case  the  discount  disappears  entirely. 
If  the  discount  is  not  taken  the  entry  would  be : 

Creditor $1 ,000 

Cash $1 ,000 

This  leaves  the  charge  of  $20  in  the  cash  discount  account  as 
an  expense. 


CHAPTER  IX 

TURNOVER 

Indefinite  Meaning  of  "  Turnover" 

The  principal  difficulty  in  discussing  turnover  is  to  obtain  a 
clear  idea  of  what  is  meant  by  the  term. 

In  spite  of  the  efiforts  of  several  committees  of  the  American 
Association  of  Public  Accountants,  we  are  no  nearer  an  authori- 
tative standard  of  accounting  terminology  than  we  were  ten  years 
ago.  The  principal  difficulty  in  arriving  at  correct  definitions  is 
that  few  authors  on  accounting  subjects  attempt  any  definitions 
at  all.  They  seem  to  take  for  granted  that  everyone  else  attaches 
the  same  meaning  to  a  term  as  they  do  themselves  and  that  a 
definition  of  it  is,  therefore,  unnecessary.  There  does  not  seem 
to  be  any  formal  definition  of  "turnover"  in  any  standard  work 
on  accounting,  and  the  word  does  not  appear  to  be  used  with  any 
clearcut  meaning. 

R.  H.  Montgomery  comes  nearest  to  defining  the  word,  but 
even  he  only  suggests  a  definition.  After  stating  that  authorities 
differ  greatly  as  to  what  the  term  means,  he  says:  "Uniformity  is 
desirable  in  accounting  terminology,  so  the  author  suggests  this 
definition:  The  turnover  of  a  merchant  or  manufacturer  repre- 
sents the  number  of  times  his  capital  in  the  form  of  stock-in- 
trade  is  re-invested  in  stock-in-trade  during  a  given  period."^ 

This  is  the  generally  accepted  definition  of  turnover;  that  is, 
the  number  of  times  the  merchandise  is  turned  over.  Mr.  Mont- 
gomery further  says:  "To  ascertain  the  turnover,  take  the  start- 
ing inventory,  add  the  purchases  or  cost  of  manufactured  goods, 
and  deduct  the  inventory  at  the  end;  divide  the  total  by  the  start- 


'  R.  H.  Montgomery,  Auditing  Theory  and  Practice,  1919,  p.  455. 

83 


84  MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 

ing  inventory.  The  result  will  be  the  number  of  times  the  capital 
invested  in  stock-in-trade  has  been  turned  over  during  the  period. 
The  calculations  are  based  upon  a  normal  inventory."^ 

Normal  Inventories  Necessary 

The  application  of  this  rule  to  two  successive  years  of  a 
business  will  exhibit  the  importance  of  the  qualification  that  the 
inventories  must  be  normal.  It  will  also  show  the  difficulty  of 
determining  in  the  case  of  the  majority  of  concerns  how  many 
times  the  stock-in-trade  is  turned  over. 

If  the  starting  inventory  of  the  first  year  is  $25,000,  the  pur- 
chases $200,000,  and  the  inventory  at  the  end  is  $50,000,  the 
turnover  during  the  year  is  7,  since  $25,000  -j-  $200,000  —  $50,000, 
or  $1 75,000,  is  seven  times  the  first  inventory.  If  in  the  following 
year  the  starting  inventory  is  $50,000,  the  purchases  $175,000, 
and  the  ending  inventory  $25,000,  the  turnover  is  4,  since  $50,000 
+  $175,000  —  $25,000,  or  $200,000,  is  four  times  the  first  inven- 
tory. Thus  by  the  mere  accident  of  a  difference  in  the  amount  of 
the  inventories  at  the  beginning  of  the  two  years,  the  second  year, 
which  did  the  larger  business,  shows  only  a  little  more  than  half 
of  the  turnover  of  the  first  year.  As  the  number  of  turnovers  in 
the  year  is  supposed  to  be  a  measure  of  the  prosperity  of  the  busi- 
ness, this  method  of  determining  it  is  evidently  unsatisfactory. 

The  difficulty  is  greatly  reduced  if  it  is  possible  to  determine 
the  average,  normal  quantity  of  stock-in-trade  carried.  This 
normal  quantity  may  or  may  not  be  the  same  as  the  inventory  at 
the  beginning  of  the  year  as  there  is  no  necessary  connection 
between  the  two.  Usually,  however,  we  only  know  that  there  was 
a  total  turnover  of  $175,000  or  $200,000  during  the  year.  Not 
knowing  the  amount  of  the  concern's  normal  inventory  from  any 
figures  contained  in  either  its  revenue  statement  or  balance  sheet, 
we  have  no  means  of  finding  how  many  times  the  stock  has  been 


^  R.  H.  Montgomery  "Auditing  Theory  and  Practice,"  1919.  P-  455- 


TURNOVER  85 

turned  over.  This  we  can  ascertain  only  it  inventories  are  taken 
monthly  or  at  other  intervals  throughout  the  year,  the  average  of 
which  may  be  regarded  as  the  normal  inventory.  In  any  event 
it  is  necessary  to  deline  the  word  "normal"  as  it  may  mean  the 
average  inventory  that  should  be  carried  or  the  inventory  that  is 
customarily  carried. 

Different  Bases  of  Comparison 

Even  if  this  difficulty  is  met,  another  presents  itself  when 
choosing  a  basis  for  comparing  the  turnovers  of  two  concerns. 
Mr.  Montgomery  probably  represents  at  least  the  majority  of 
American  accountants  in  considering  turnover  to  be  the  number 
of  times  the  normal  stock-in-trade  is  reinvested  in  the  goods  sold. 
British  accountants,  on  the  other  hand,  following  Lisle,  say  that 
it  should  be  the  relation  between  the  inventory  and  the  sales.  It 
can  readily  be  seen  that  a  serious  misunderstanding  results  in  the 
comparison  of  the  turnovers  of  two  concerns  if  one  is  calculated 
on  the  basis  of  cost  and  the  other  on  the  basis  of  sales,  especially 
if  high  selling  expenses  make  the  cost  of  the  goods  a  compara- 
tively small  part  of  the  selling  price. 

This  difference  of  opinion  as  to  the  proper  basis  for  comparing 
turnovers  has  reference  to  trading  businesses.  In  the  case  of 
manufacturing  businesses  the  disagreement  is  even  greater,  as 
three  bases  of  comparison  may  be  used,  raw  material,  raw  ma- 
terial plus  labor,  and  the  total  cost  of  the  goods  sold. 

Working  Capital  as  Basis  of  Turnover 

Owing  to  this  wide  divergence  of  opinion,  it  has  been  sug- 
gested, and  it  would  seem  justifiably,  that  a  better  basis  for  cal- 
culating the  turnover  is  the  working  capital  of  the  business.  If 
this  method  is  adopted,  most  of  the  difficulties  encountered  in 
computing  the  turnover  disappear,  because  the  amount  of  work- 
ing capital  is  indicated  in  the  balance  sheet,  being  the  excess  of 
current  assets  over  current  liabilities.    Except  as  affected  by  the 


86  MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 

slight  increase  due  to  undistributed  profits,  it  remains  the  same 
throughout  the  entire  period,  and,  therefore,  affords  a  stable  basis 
of  comparison. 

Every  business,  whether  it  is  engaged  in  trading  or  manufac- 
turing, requires  capital  for  two  purposes:  first,  to  provide  the 
necessary  fixed  assets  to  carry  on  the  operations;  and  second,  to 
furnish  sufficient  funds  to  carry  the  stock-in-trade  and  accounts 
receivable  until  cash  is  realized  on  them  and  more  goods  or  mate- 
rial for  manufacture  are  bought.  The  point  that  interests  the 
proprietor  is  how  many  times  a  year  he  can  invest  his  available 
floating  capital  by  repeating  the  process  of  buying  the  goods, 
selling  them,  and  collecting  the  proceeds.  If  one  person  with  a 
working  capital  of  $40,000  is  able  to  sell  in  a  year  goods  costing 
$200,000,  while  another  with  the  same  amount  of  working  capital 
is  able  to  sell  goods  costing  only  $160,000,  the  first  person  has 
turned  over  his  capital  five  times  to  the  other  person's  four  times. 

Some  exception  is  taken  to  this  definition  of  turnover.  It  is 
said  that  working  capital  is  used  for  many  purposes,  including 
advertising  campaigns  and  similar  items  of  expense  that  do  not 
affect  the  cost  of  manufacture.  It  is  also  objected  that  accounts 
receivable  vary  to  a  great  extent  between  different  concerns,  as 
some  sell  on  short  time  and  others  on  long  time,  and  some  keep  a 
big  stock  of  raw  materials,  while  others  do  not  consider  it  neces- 
sary, or  cannot  afford  to  do  so. 

The  question  of  what  constitutes  the  proper  basis  for  reckon- 
ing the  turnover  seems  to  turn  on  the  object  for  which  the  turn- 
over is  used.  If  it  is  merely  for  the  purpose  of  showing  that  one 
manager  can  handle  more  goods  than  another  with  the  same 
average  amount  of  stock  on  hand,  the  proper  test  is  the  relation 
between  the  normal  inventory  and  the  cost  of  the  goods  sold. 
But  a  manager  could  establish  a  record  on  this  basis  by  following 
the  foohsh  policy  of  buying  in  small  quantities  at  retail  prices  and 
paying  the  high  expressage  instead  of  the  low  freight  rates.  His 
turnover  would  be  large,  but  would  lead  to  disaster. 


TURNOVER  87 

If  a  profitable  business  is  the  object  sought  for,  the  use  the 
manager  makes  of  the  working  capital  would  seem  to  be  one  of  the 
best  measures  of  his  success.  If  too  much  of  the  working  capital 
is  diverted  from  the  production  or  purchase  of  goods  for  sale  to 
carrying  on  an  extensive  advertising  campaign,  or  if  it  is  tempora- 
rily locked  up  in  long-time  accounts  receivable,  the  quantity  of 
goods  sold  is  apt  to  decrease.  This  will  show  itself  in  a  lessened 
turnover  of  working  capital.  On  the  other  hand,  if  either  the 
beginning  or  the  normal  inventory  is  made  the  measure,  the 
turnover  will  remain  the  same  even  though  the  business  done  is 
smaller,  because  the  inventory  will  decline  with  the  volume  of 
business.  An  inventory  of  $25,000  and  cost  of  sales  totaling 
$100,000  will  show  the  same  turnover  on  this  basis  as  $50,000  in- 
ventory and  cost  of  sales  totaling  $200,000.  When  the  yardstick 
varies  in  length,  comparative  measurements  are  of  little  value. 

The  use  of  working  capital  as  the  basis  of  turnover  is  logical, 
first,  because  the  capital  is  put  in  the  business  for  the  purpose 
of  being  turned  over  as  rapidly  as  possible;  second,  because  it  is 
virtually  constant;  and  third,  because  it  presents  all  the  elements 
concerned  in  the  turnover,  not  only  the  stock-in-trade,  but  also 
the  accounts  and  notes  receivable,  by  means  of  which  the  turnover 
is  effected.  The  turnover  of  working  capital  also  furnishes  a 
better  criterion  of  the  excellence  of  the  management.  With  the 
inventory  as  the  only  standard  a  manager  can  make  an  apparently 
good  record  by  starving  his  stock-in-trade.  If,  however,  he  uses 
working  capital  as  the  standard,  he  makes  his  best  record  by 
dihgence  in  collecting  outstanding  accounts,  and  increasing 
the  supply  of  cash  for  the  development  and  handling  of  a  more 
extensive  business. 

Definition  of  Working  Capital 

Even  if  this  is  granted,  the  difficulty  does  not  come  to  an  end, 

because  there  is  no  authoritative  detinition  of  working  capital. 

H.  R.  Hatfield  says:  "Working  capital  has  long  had  a  specific 


88  MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 

meaning  as  a  collective  term  for  what  are  often  called  quick 
assets,  e.  g.,  cash,  accounts  receivable,  perhaps  merchandise,  etc."  ^ 

H.  C.  Bentley  says:  "Working  capital  is  the  excess  of  quick 
assets  over  quick  liabilities."'* 

Each  of  these  authorities  has  his  followers,  as  was  shown  in  a 
recent  discussion  of  the  subject  in  the  Journal  of  Accountancy. 
The  definitions  differ  because  in  one  the  notes  and  accounts  pay- 
able are  considered  to  be  borrowed  capital,  while  in  the  other 
only  the  amount  contributed  by  the  proprietor  is  treated  as  capi- 
tal. The  first  is  the  economic  view  of  what  constitutes  capital, 
while  the  second  is  the  business  and  general  accounting  view. 
If  the  ordinary  business  man  is  asked  how  much  capital  he  has  in 
his  business,  he  will  always  state  the  amount  of  his  proprietary 
interest.  Unlike  Micawber,  he  does  not  think  he  has  added  to  his 
capital  whenever  he  issues  a  note  payable. 

Need  of  Exact  Definitions 

The  final  lesson  to  be  learned  from  the  consideration  of  this 
subject  is  that  in  preparing  comparative  statements  of  turnover, 
accountants  should  first  define  the  terms  used  and  should  plainly 
state  the  basis  of  the  calculations.  Otherwise  the  conclusions 
reached  will  be  entirely  misleading  to  persons  whose  conception 
of  the  subject  is  dift'erent.  It  is  to  be  hoped  that  the  Institute  of 
Accountants  will  eventually  end  the  present  ambiguity  by  formu- 
lating an  authoritative  definition  of  turnover. 

Having  settled  upon  the  basis  to  be  used,  the  accountant  de- 
termines the  turnover  by  ascertaining  how  many  times  the  normal 
inventory  will  go  into  the  cost  of  the  goods  sold,  or  into  the  sales, 
according  to  which  view  is  adopted,  or  how  many  times  the  work- 
ing capital  has  been  reinvested  in  the  purchase  or  production  of 
goods  sold.  The  greater  the  quotient  in  each  case,  the  more  pros- 
perous and  better  managed  the  business  is  supposed  to  be. 


3  H.  R.  Hatfield,  Modern  Accounting,  1909,  p.  179. 

4  H.  C.  Bentley,  Science  of  Accounts,  191 1. 


CHAPTER   X 

PARTNERSHIPS 

Division  of  Profits 

Profits  may  be  divided  by  partners  in  any  proportions  to 
which  they  agree;  if  they  malce  no  express  agreement  the  law  im- 
plies an  agreement  to  divide  the  profits  equally,  regardless  of  the 
capital  or  services  contributed.  The  customary  methods  of 
dividing  profits  are: 

1.  In  the  ratio  of  the  capital  balances  at  the  beginning  of  the 

period. 

2.  In  the  ratio  of  the  average  capitals  for  the  period. 

3.  In  an  arbitrary  ratio,  usually  expressed  in  terms  of  frac- 

tions or  per  cents. 

4 .  In  an  arbitrary  ratio  after  allowing  interest  on  the  capitals. 

To  illustrate  these  methods,  assume  the  following  facts: 

Illustration 

A  and  B  are  in  partnership  and  the  profits  for  division  at  the  end  of 
the  year  are  $12,000. 

A's  capital  account  during  the  year  undergoes  the  following  changes: 

Credit  balance,  January  i $50,000 

Investment,  March  i 2,000 

"  November  i 3,000 

Total  credits $55,000 

Withdrawal,  May  i $500 

"  December  i 1,000 

Total  debits 1,500 

Credit  balance,  December  31 $53,500 


90 


MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 


B's  capital  account  changes  as  follows: 

Credit  balance,  January  i $25,000 

Investment,  February  i 10,000 

"  June  1 5,000 

Total  credits 

Withdrawal,  October  i 

Credit  balance,  December  31 


j.0,000 
1,000 

59, 000 


Solution  i:  Division  of  profits  in  the  ratio  of  the  capital  balances  at  the 
beginning  of  the  year: 

This  ratio  is  A,  50,  and  B,  25;  or  2  to  i.  A,  therefore,  is  credited  with 
%  of  $12,000,  or  |8,ooo;  and  B  with  }4  of  $12,000,  or  $4,000. 

Solution  2:  Division  of  profits  in  the  ratio  of  the  average  capital  for  the 
period: 

This  ratio  may  be  computed  in  either  of  two  ways.  The  first  is  as 
follows:  Multiply  each  capital  account  credit  by  the  number  of  months 
or  days  from  the  date  of  the  credit  until  the  end  of  the  period,  and  find 
the  sum  of  these  products.  Multiply  each  capital  account  debit  by  the 
number  of  months  or  days  from  the  date  of  the  debit  until  the  end  of 
the  period,  and  find  the  sum  of  these  products.  Find  the  difference 
between  the  credit  products  and  the  debit  products. 

Do  this  with  each  capital  account  and  determine  what  fraction  each 
difference  is  of  the  sum  of  the  differences.  The  difference  between  the 
credit  and  debit  products  of  A's  capital  account  is  found  thus: 


Credits 

Date  Amount  Time 

January      i $50,000  12  mo. 

March         i 2,000  10    " 

November  i 3,000  2     " 

Debits 

May  I $     500       8  mo. 

December  i i  ,000       r     " 

Difference 


Product 

$600,000 

20,000 

6,000 

$626,000 

$  4,000 

1,000 

5,000 

$621,000 

PARTNERSHIPS 


91 


The  difference  between  the  credit  and  debit  products  of  B's  capital  account 
is  found  thus: 


Credits 

Date  Amount     Time 

January      i $25,000     12  mo. 

February    i 10,000     11     " 

June  1 5.000       7     " 

Debits 
October      i $1,000        3     mo. 

Difference 


Product 

$300,000 
110,000 
35,000      $445, 00c 


3,000 


|.2,000 


The  partnership  profits  of  $12,000  are  accordingly  divided  in  the  follow- 
ing ratios  and  amounts: 


Ratios  Expressed  in  Fractions 

A 621/1063 

B 442/1063 


Division  of  Profits 

$  7,010.35 
4,989.65 


$12,000.00 


The  second  method  by  which  profits  may  be  divided  in  the  ratio  of  the 
average  capitals  for  the  period  is  as  follows: 

Multiply  the  opening  balance  of  each  account  by  the  number  of 
months  or  days  it  remained  unchanged. 

Multiply  each  new  balance  resulting  from  investments  or  withdrawals 
by  the  number  of  months  or  daj's  it  remained  unchanged. 

Find  the  sum  of  these  products  for  each  capital  account. 

Find  the  ratio  of  each  sum  to  the  total  for  all. 

The  products  for  A's  capital  account  are  computed  in  the  following 
way: 


92 


MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 


Balance 

From 

To 

Time 

Product 

$50,000 

January 

March 

2  mos. 

$100,000 

52,000 

March 

May 

2    " 

104,000 

51,500 

May 

November 

6    " 

309,000 

54,500 

November 

December 

I  mo. 

54,500 

53,500 

December 

December 

31 

I     " 
12  mos. 

53,500 

$621,000 

The  products  for  B's  capital  account  are  figured  as  follows: 


Balance 

From 

To 

Time 

Product 

$25,000 

January       i 

February 

I 

I  mo. 

$  25,000 

35'000 

February     i 

June 

I 

4  mos. 

140,000 

40,000 

June            I 

October 

I 

4     " 

160,000 

39,000 

October       i 

December 

31 

3     " 

117,000 

This  method  results  in  the  same  ratios  as  the  first  method,  and  it  has  the 
advantage  that  the  final  balances  shown  in  the  computation  ($53,500  and 
$39,000)  are  the  same  as  the  balances  of  the  accounts;  and  that  the  time 
numbers  used  as  multipliers  add  to  a  full  year.  Checks  on  the  accuracy  of 
the  computations  are  thus  provided. 

It  must  be  understood  that  these  computations  determine  the  average 
capital  ratios  but  not  the  average  capitals.  To  compute  the  average 
capitals  it  would  be  necessary  to  divide  by  12,  thus: 

$621,000  -^  12  =  $51,750.00,  A's  average  capital 
$442,000  H-  12  =  $36,833.33,  B's         " 

Since  only  the  ratio  between  the  average  capitals  is  required,  the  division 
by  12,  or  by  365  if  the  numbers  of  days  have  been  used  as  multipliers,  is 
unnecessary. 

Solution  3 :  Division  of  profits  in  an  arbitrary  ratio.  No  limit  can 
be  placed  on  the  variety  of  arbitrary  ratios  which  can  be  agreed  upon 
and  illustrations  are  unnecessary. 

Solution  4:  Division  of  profits  in  an  arbitrary  ratio  after  allowing 
interest  on  capital.  No  interest  can  be  allowed  unless  there  is  a  specific 
agreement  to  do  so,  and  in  that  event  the  rate  should  be  agreed  upon. 


PARTNERSHIPS  93 

Assuming  a  rate  of  6%  on  the  opening  balances  of  the  capital  accounts, 
the  distribution,  if  the  remaining  profits  are  divided  equally,  would  be  as 
follows : 

A  B  Total 

6%  of  $50,000 $3,000 

6%  of    25,000 $1,500 

Total  interest $4,500 

Balance  equally 3,750       3,750         7,500 

$6,750     $5,250     $12,000 


If  the  agreement  provides  for  interest  on  partners'  capitals,  it  must 
be  credited  to  them  even  though  it  exceeds  the  total  profits.  In  that 
event  the  resulting  debit  balance  in  the  profit  and  loss  account  is  charged 
to  the  partners  in  the  agreed  ratio. 

Assuming  that  the  profits  were  only  $4,000,  the  division  would  be  as 
follows: 

A  B         T0T.VL 

Credits  for  interest  (as  above) $3,000     $1,500     $4,500 

Debits  for  excess  of  interest  over  profits         250  250  500 

Net  credits $2,750     $1,250     $4,000 

If  interest  is  provided  for  in  the  agreement,  it  must  be  credited  to  the 
partners  even  though  a  loss  has  been  incurred  instead  of  a  profit  earned. 
Assuming  a  loss  of  $1,000,  the  division  would  be  as  follows: 

A  B  Total 
Debits  for  sum  of  loss  of  $1 ,000  and  in- 
terest of  $4,500 $2,750  $2,750  $5,500 

Credits  for  interest 3,000  1,500  4,500 

Net  credit $250 

Net  debit $1,2^0     $1,000 


The  result  is  that  B  bears  all  of  the  loss  from  operations  as  well  as  the 
$250  net  credit  to  A. 


94  MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 

Liquidation  of  Partnerships 

When  a  partnership  is  terminated,  the  procedure  to  be  fol- 
lowed in  realizing  the  assets,  liquidating  the  liabilities  and  dis- 
tributing the  partners'  capitals,  depends  on  whether  all  losses  on 
realization  have  been  ascertained  before  payments  are  made  to 
the  partners.  If  they  have  been,  the  losses  are  deducted  from  the 
partners'  capitals  in  the  profit  and  loss  ratio,  and  the  remaining 
assets,  after  paying  the  outside  creditors,  are  distributed  to  the 
partners  in  amounts  sufficient  to  pay  off  the  capitals. 

To  illustrate,  assume  that  all  partnership  debts  have  been  paid, 
and  that  the  capital  accounts  are: 

A  $10,000        B  $8,000 

There  must  be  assets  of  $18,000.  These  are  sold  for  $15,000. 
With  losses  divided  equally,  the  division  of  cash  proceeds  is  as 

follows : 

A              B  Total 

Capitals $10,000  $8,000  $18,000 

Losses  on  realization i,Soo       1,500  3,000 

Balances  paid  in  cash $8,500     $6,500     $15,000 


Periodical  Distributions 

If  periodical  distributions  to  the  partners  are  made  before  all 
assets  are  realized  and  all  losses  ascertained,  they  should  be 
made,  if  possible,  in  such  a  way  as  to  reduce  the  balances  of  the 
capital  accounts  to  the  profit  and  loss  ratio  existing  between  the 
partners,  so  that  if  all  remaining  assets  are  lost  each  partner's 
capital  account  will  be  exactly  sufficient  to  cover  his  share  of  the 
loss. 

To  illustrate,  assume  that  all  liabilities  are  paid  and  the  part- 
ners' capitals  are  as  follows : 

A  $15,000 
B  20,000 
C     25,000 


PARTNERSHIPS  95 

The  assets  total  $60,000.  In  realizing  on  $i8,oco  worth  of 
assets  a  loss  of  $3,000  is  incurred,  so  that  there  is  $15,000  in  cash 
to  divide.  The  division  of  cash  should  be  made  as  follows,  as- 
suming that  the  partners  share  profits  and  losses  equally. 

ABC  Total 

Capitals $15,000     $20,000     $25,000     $60,000 

Loss 1,000         1,000         1,000        3,000 

Balance  before  dividing  cash $14,000     $19,000     $24,000     $57,000 

Cash 5.000       10,000       15,000 

Balances  left  in  P.  &  L.  ratio $14,000     $14,000     $14,000     $42,000 


It  should  be  noted  that  the  cash  is  not  distributed  in  the  capi- 
tal ratio,  the  profit  and  loss  ratio,  nor  any  other  ratio,  but  in 
arbitrary  amounts  sufficient  to  reduce  the  capitals  to  the  profit 
and  loss  ratio. 

Reducing  Capitals  to  Profit  and  Loss  Ratio 

It  is  not  always  possible  to  bring  the  capital  account  balances 
to  the  profit  and  loss  ratio  at  the  first  distribution  of  cash.  This 
is  the  case  if  the  capital  of  any  partner  after  charging  off  all  ascer- 
tained losses  is  less  than  his  profit  and  loss  ratio  of  the  assets 
which  will  remain  after  making  the  proposed  distribution. 

To  illustrate,  assume  the  capitals  to  be  as  follows: 

A  $10,000 

B  20,000 

C  30,000 

Losses  are  to  be  shared  equally.  The  assets  total  $60,000  and  all 
Habilities  are  paid.  Assets  carried  on  the  books  at  $30,000  are 
sold  for  $24,000,  the  loss  of  $6,000  being  divided  as  follows: 


ABC 

Total 

Capitals  before  dividing  loss 

Loss 

Capitals  before  distribution 
of  cash  

$10,000     $20,000     $30,000 
2,000         2,000         2,000 

$60,000 
6,000 

$8,000     $18,000     $28,000 

$54,000 

96  MATHEMATICS  OP  ACCOUNTING  AND  FINANCE 

After  the  $24,000  is  divided  there  will  remain  a  total  capital  of 
$30,000.  If  possible  the  $24,000  cash  should  be  divided  in  such  a 
way  as  to  leave  each  partner  with  a  balance  of  $10,000;  but  this 
is  clearly  impossible  since  A's  balance  is  already  reduced  to 
$8,000.  Since  A's  capital  is  not  sufficient  to  bear  his  share  of  the 
total  possible  loss  of  the  $30,000  of  assets  which  remain  after  the 
distribution,  nothing  should  be  paid  to  A.  In  the  event  that  a 
total  loss  of  $30,000  is  incurred  the  charge  of  $10,000  to  A  would 
leave  his  account  with  a  debit  balance  of  $2,000.  If  he  could  not 
pay  in  the  $2,000  it  would  have  to  be  charged  against  B  and  C 
in  their  profit  and  loss  ratios,  which  in  this  case  happen  to  be 
equal.  Therefore,  B  and  C  may  possibly  lose  the  following 
amounts : 

B  C 

One-third  each  of  total  possible  $30,000  loss     $10,000  $10,000 
Excess  of  A's  share  of  possible  loss  over  his 

capital 1 ,000  1 ,000 


Total  possible  loss $11,000     $11,000 


The  $24,000  cash  should  be  divided  between  the  two  partners 
in  such  a  way  as  to  reduce  their  balances  to  these  amounts,  as 
follows : 

ABC  Total 
Capitals     before  distribu- 
tion of  cash $8,000     $r8,ooo     $28,000  $54,000 

Cash  distributed 7,000       17,000  24,000 


Balances $8,000    $11,000    $11,000    $30,000 

In  the  next  realization,  assets  carried  at  $15,000  are  realized 
at  $12,000,  the  loss  being  $3,000.  After  dividing  the  loss  the 
cash  can  be  distributed  in  amounts  which  will  reduce  the  capital 
balances  to  the  profit  and  loss  ratio  of  equality. 


PARTNERSHIPS 

ABC  Total 

Balances  (as  above) $8,000  $11,000  $11,000  $30,000 

°^^ i-ooo  1,000  1,000  3,000 

Balances  before  distribut- 

^"S  cash $7,000  $10,000  $10,000  $27,000 

^^    2,000  5,000  5,000  12,000 

Balances  (reduced  to  P.  & 

^-  ^^^^°) $5,000     $  5,000       $5,000     $15,000 

The  remaining  assets  are 
sold  for  $9,000,  the  di- 
vision of  loss  and  cash 
being  as  follows: 

^°^^ 2,000         2,000         2,000         6,000 

Balances  paid  in  cash ...  .     $3,000    $3,000    $  3,000    $  9,000 


97 


CHAPTER  XI 

THE  CLEARING  HOUSE 

Principle  of  the  Clearing  House 

The  general  principle  on  which  a  clearing  house  operates  is 
that  of  offsetting  debits  and  credits  and  dealing  only  with  net 
differences.  Its  simplest  form  is  illustrated  when  two  persons  buy 
from  and  sell  to  each  other.  If  instead  of  each  paying  his  bill  to 
the  other  in  full,  the  one  who  owes  the  larger  sum  deducts  the 
other's  debt  to  him  and  pays  the  difference,  he  has  to  that  extent 
adopted  the  clearing  house  principle. 

In  the  fullest  application  of  the  principle,  an  outside  party  is 
introduced  to  act  as  settling  or  clearing  agent.  Each  unit  in  the 
combination  that  forms  the  clearing  house  has  dealings  with  every 
other  unit,  but  instead  of  settling  its  dealings  with  each  of  the 
other  units  individually  it  charges  the  total  of  all  its  debits  and 
credits  the  total  of  all  its  credits  to  the  clearing  agent,  with  which 
it  settles  the  net  difference  of  the  totals.  In  this  way  a  single 
settlement  takes  the  place  of  a  large  number  of  settlements.  An 
example  will  make  this  clear. 

Debits  and  Credits  with  Clearing  House 

Suppose  there  are  six  banks  in  a  city.  Every  day  they  re- 
ceive on  deposit  and  otherwise,  checks  on  each  other.  If  there 
is  no  clearing  house,  each  bank  presents  to  each  of  the  others 
the  checks  drawn  on  it  and  collects  or  pays  the  difference  due 
to  or  from  it,  according  as  the  amount  of  the  checks  it  pre- 
sents is  greater  or  smaller  than  the  amount  of  those  presented 
to  it. 

If  the  banks  form  a  clearing  house,  each  provides  itself  with  a 
blank  form,  in  which  it  inserts  the  amount  of  checks  it  holds 

98 


THE  CLEARING  HOUSE 


99 


against  each  of  the  others.  The  total  it  charges  to  clearing 
house  checks  on  the  teller's  blotter.  This  form  and  the  checks 
themselves  are  sent  to  the  clearing  house.  The  checks  are  de- 
livered to  the  representatives  of  the  banks  on  which  they  are 
drawn,  with  a  memorandum  of  the  amount  for  each  bank.  As 
each  bank  receives  the  checks  drawn  on  it,  it  enters  the  amounts 
in  the  second  column  of  the  form  and  the  total  becomes  the  credit 
to  clearing  house  checks.  If  a  bank's  credits  to  the  clearing  house 
are  greater  than  its  debits,  it  pays  the  difference  to  the  clearing 
house.  The  amounts  thus  paid  in  are  afterwards  paid  to  those 
banks  whose  debits  to  the  clearing  house  are  greater  than  their 
credits. 

Clearing  House  Transactions 

The  following  are  assumed  to  be  the  transactions  on  a  certain 
day  of  the  six  banks  in  the  clearing  house,  it  being  remembered 
that  the  first  column  of  figures  on  each  blank  is  made  up  before 
the  checks  leave  the  bank  and  the  second  column  at  the  clearing 
house. 


First  National  Bank 

Atlas  Nat ^27,819.32 

Merchants  St 12,948.24 

Traders  St 22,687.19 

State  Tr.  Co 18,729.63 

Union  Nat 14,963.48 


Atlas  National  Bank 

$16,248.76  First  Nat $16,248.76 

14,629.83  Merchants  St 11,283.42 

26,963.42  Traders  St 12,732.63 

21,246.38  State  Tr.  Co 15,376.28 

IS. 782. 16  Union  Nat 9.659.87 


C.  H.  owes . 


197,147-86     $94,870.55 

$   2,277-31     Owes  C.  H. 


$27,819.32 

8,664.19 

10,497.26 

11.958.47 

7,326.58 


$65,300.96     $66,265.82 
$      964.86 


Merchants  State  Bank 


i^irst  Nat 

. .     $14,629.83 

$12,948.24 

Atlas  Nat 

8.664.19 

11,283.42 

Traders  St 

9.586.28 

10,392.16 

State  Tr.  Co  .  .  . 

11,473.47 

9.627.59 

Union  Nat 

10,827.42 

11.432.27 

$55,181.19 

$55,683.68 

Owes  C.  H 

$      502. 49 

Traders  State  Bank 

First  Nat $26,963.42 

Atlas  Nat 10,497.26 

Merchants  St 10,392.16 

State  Tr.  Co 11,788.39 

Union  Nat 9.673.57 


C.  H. 


$22,687.19 
12,732.63 

9,586.28 
13.249.36 

8,865.71 


$69,314-80     $67,121.17 
$  2,193.63 


100         MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 


s 

First  Nat . 

TATE 

Trust  Compan 

Y 

$18,729.63 
15,376.28 
11.473-47 
11,788.39 

8,842.74 

Union 

First  Nat 

Atlas  Nat 

National  Ban 

. .     $15,782-16 

7,326.58 

11,432.27 

8,865.71 

8,842.74 

K 

Atlas  Nat . 

St... 

11,958.47 
9.627.59 
13,249.36 

7.982.25 

9,659.87 

Merchants 
Traders  St. 
Union  Nat . 

Merchants  St.  .  . 

Traders  St 

State  Tr.  Co 

Owes  C.  H 

10,827-42 
9.673.57 
7,982.2s 

Owes  C.  H. 

$64,064.05 
$  2,146.46 

$66,210.51 

$52,249.46 
. .      $      857-13 

$53,106.59 

Debit 

Credit 

Debits 

Credits 

Balances 

Balances 

$  94.870.SS 

$  97.147.86 

$ 

$2,277-31 

66,265.82 

65.300.96 

964.86 

55.683-68 

SS,l8l-I9 

502.49 

67.121. 17 

69.314-80 

2,193.63 

66,210.51 

64.064.05 

2,146.46 

S3. 106.59 

52,249-46 

857.13 

$403,258.32 

$403,258.32 

$4,470.94 

$4,470-94 

Manager's  Sheet 

The  representatives  of  the  different  banks  are  allowed  a  cer- 
tain number  of  minutes  in  which  to  finish  the  preparation  of  their 
respective  sheets.  They  then  in  turn  call  out  their  figures  which 
are  registered  by  the  manager  of  the  clearing  house  on  his  own 
sheet  in  the  following  manner: 


Banks 

First  National  Bank 
Atlas  National  Bank 
Merchants  State  Bank 
Traders  State  Bank 
State  Trust  Company 
Union  National  Bank 


If  the  manager's  sheet  balances,  he  rings  his  bell  to  signify 
that  the  session  is  over.  If  it  does  not  balance  he  announces  the 
fact.  A  certain  number  of  minutes  is  allowed  for  finding  the 
error.  If  found  within  that  time  the  one  who  made  the  error  is 
fined  a  certain  sum.  The  fine  increases  progressively  with  any 
added  time  taken  to  reach  a  balance. 

Later,  the  banks  with  debit  balances  make  their  payments  to 
the  clearing  house  manager,  who  shortly  after  pays  off  the  banks 
with  credit  balances. 

The  total  figures  are  kept  by  the  manager  for  statistical  pur- 
poses. The  sum  of  the  checks  presented  ($403,258.32)  is  the 
clearing  house  movement  for  the  day,  and  the  sum  of  the  balances 
($4,470.94)  is  the  amount  of  the  balances  reported  in  the  financial 


THE  CLEARING  HOUSE  lOI 

news.  The  corresponding  figures  for  the  week,  month  and  year 
are  published  periodically  in  the  financial  columns  of  the  news- 
papers, and  are  considered  a  barometer  of  business  activity. 

Economy  of  System 

The  convenience  and  the  economy  of  the  clearing  house  sys- 
tem may  be  demonstrated  by  an  analysis  of  the  transactions  of 
the  Merchants  State  Bank.  This  bank  has  collected  checks 
aggregating  $55,181.19  and  has  paid  checks  totaling  $55,683.68, 
or  a  total  settlement  of  $1 10,864.87,  by  a  payment  of  only  $502.49, 
or  less  than  one  half  of  1%.  Without  the  clearing  house,  it 
would  have  been  compelled,  even  by  offsetting  with  each  of  the 
other  banks,  to  make  the  following  payments : 

Atlas  National  Bank $2,619.23 

Traders  State  Bank 805.88 

Union  National  Bank 604.85     $4,029.96 

It  would  have  received  the  following  amounts: 

First  National  Bank $1,681.59 

State  Trust  Company 1,845.88       3,527.47 

Total  cash  movement  in  the  bank  would  have  been. . .     $7,557.43 


One  payment  of  $502.49  takes  the  place  of  five  transactions 
aggregating  fifteen  times  as  much. 

Each  bank  must  settle  on  the  basis  of  the  clearing  house  re- 
turns. The  corrections  of  any  errors  in  addition,  the  listing  of 
checks,  either  for  wrong  amounts  or  on  the  wrong  bank,  etc.,  and 
the  settlement  for  checks  returned  for  lack  of  funds  or  on  account 
of  missing  endorsements,  are  all  matters  left  to  the  individual 
banks  to  adjust  with  each  other. 

Application  of  Principle  Extended 

The  clearing  house  principle  is  applied  to  the  settlement  of 
accounts  whenever  several  parties  have  reciprocal  relations,  in- 


102         MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 

volving  the  transfer  of  value  in  the  form  of  money,  securities,  or 
merchandise.  It  is  applied  to  the  settlement  of  accounts  between 
brokers  on  boards  of  trade  and  other  exchanges,  and  also  to  the 
regulation  of  sales  of  merchandise  when  an  agreement  exists 
among  concerns  in  the  same  line  of  business  restricting  the  output 
of  each  to  an  agreed  percentage.  At  present  there  is  very  little 
of  the  latter  use  of  the  method,  owing  to  the  danger  of  violating 
the  law  which  prohibits  agreements  in  restraint  of  trade. 


CHAPTER  XII 

BUILDING  AND  LOAN  ASSOCIATIONS 

General  Characteristics  of  Building  and  Loan  Associations 

A  building  and  loan  association  is  founded  on  the  general 
principle  of  co-operation  among  a  large  number  of  small  investors 
and  a  smaller  number  of  intending  borrowers,  who  wish  to  get 
advances  for  the  purpose  of  erecting  buildings  on  land  they  own, 
or  of  paying  off  existing  loans  on  real  estate  that  has  been  partially 
paid  for.  There  are  several  different  plans  on  which  such  associa- 
tions are  organized,  but  the  idea  of  co-operation  is  present  in 
them  all. 

In  some  foreign  countries,  notably  Italy,  this  principle  of  co- 
operation has  been  extended  to  cover  loans  to  workmen  for  the 
purchase  of  tools  or  machinery  and  even  to  the  furnishing  of 
large  sums  to  working  contractors  for  building  railroads  and  other 
public  works. 

The  usual  characteristics  of  a  building  and  loan  association 
are  as  follows:  Each  member  subscribes  to  a  definite  number  of 
shares,  on  which  he  pays  monthly  instalments  of  one  half  of  i% 
of  the  par  of  the  stock.  If,  for  example,  the  par  is  fixed  at  $ioo, 
the  monthly  payment  on  each  share  is  50  cents.  When  these 
instalments  together  with  the  profits  earned  by  each  share 
amount  to  $100,  the  stock  is  said  to  have  matured  and  is  paid 
off. 

Terminating  Plan 

The  terminating  plan  of  organization,  the  first  to  be  adopted,  is 
the  simplest  of  all.  It  involves  the  subscription  by  a  limited  num- 
ber of  persons  to  a  certain  number  of  shares  which  are  to  be  paid 
in  instalments,  and  the  dissolution  of  the  association  upon  the 

103 


I04         MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 

maturity  of  the  shares.  As  money  is  paid  by  subscribers  it  is 
loaned  from  time  to  time.  As  the  interest  collected  is  also  loaned, 
the  association  is  in  theory  receiving  compound  interest.  For 
instance,  if  2,000  shares  are  taken  and  50  cents  per  share  is 
paid  monthly,  a  loan  of  $1,000,  say  at  6%,  can  be  made  at  the 
end  of  the  first  month.  At  the  close  of  the  second  month  there  is 
another  $1,000  to  lend,  together  with  $5  interest  collected  on  the 
first  loan,  and  this  is  repeated  from  month  to  month.  The  com- 
pound interest  theory,  however,  does  not  hold  strictly  here,  as  it 
is  difficult  to  lend  odd  amounts,  unless  other  security  than  real 
estate  is  taken. 

The  objection  to  this  plan  is  that  it  is  impossible  to  employ  the 
money  profitably  in  real  estate  loans,  because  during  the  latter 
part  of  the  association's  life  loans  have  to  be  made  for  so  short 
a  time  that  great  difficulty  is  experienced  in  making  them; 
and  if  the  later  loans  should  be  made  for  the  usual  term  of  five 
years  the  dues  and  profits  on  the  stock  could  not  be  paid 
when  they  matured.  It  can,  however,  be  arranged  that  after 
the  stock  reaches  par  no  further  payments  should  be  made  on 
the  shares  and  no  shares  should  be  withdrawn,  but  the  money 
should  be  kept  invested  and  cash  dividends  paid  out  of  the 
interest  collected. 

Practicability  of  Plan 

The  terminating  plan  can  be  used  to  best  advantage  when  a 
large  number  of  persons  wish  to  pay  in  a  given  time  a  large  sum 
of  money  with  interest,  each  person  paying  a  small  amount 
monthly  for  the  given  time.  This  plan,  for  example,  is  extremely 
useful  in  paying  off  a  church  debt.  If  a  church  without  any  rich 
members  owes  $10,000  on  which  it  is  paying  6%  interest,  it  might 
seem  impossible  to  raise  so  large  an  amount.  If  a  building  and 
loan  association  is  formed  with  100  shares  of  $100  each,  to  run  for 
five  years,  on  which  the  monthly  payments  are  fixed  at  $1.93  per 
month,  it  will  usually  not  be  found  difficult  to  persuade  the  con 


BUILDING  AND  LOAN  ASSOCIATIONS  I05 

gregation  to  take  the  whole  100  shares.  This  will  not  only  pay 
off  the  entire  debt  in  five  years,  but  will  also  pay  the  interest  on 
the  diminishing  amount  of  the  note,  as  the  holder  of  the  note 
will  be  willing  to  accept  partial  payments  of  even  hundreds  of 
dollars. 

The  plan  works  out  in  this  way — Each  holder  of  a  share  agrees 
to  pay  $100  and  interest  in  monthly  instalments  of  $1.93.  If  he 
makes  his  payments  regularly,  his  account  for  the  first  two 
months  runs  as  follows : 

Original  agreement $100.00 

One  month's  interest  at  6% .50 

$100.50 
First  month's  payment 1.93 

$  98.57 
One  month's  interest  at  6% .49 

$  99.06 
Second  month's  payment 1.93 

$  07-13 

The  treasurer  credits  the  interest  account  each  month  with 
the  interest  paid  that  month,  and  the  principal  with  the  balance 
of  the  payments,  applying  the  money  to  the  payment  of  the  inter- 
est and  principal  of  the  note. 

The  amount  of  the  monthly  payments  theoretically  necessary 
is  found  from  the  compound  interest  table  headed,  "Amount  per 
annum  necessary  to  pay  a  debt  of  $1  now  due  and  the  interest 
thereon."  In  order  to  reduce  it  to  monthly  payments  it  is  neces- 
sary to  use  the  table  of  ^  %  for  the  number  of  months  in 
the  life  of  the  association.  In  our  illustration  we  find  the  amount 
necessary,  or  $1.93,  in  the  Yi  %  column  on  the  line  of  the 
60th  period.  For  six  years  or  72  months  the  amount  is  $1.65^. 
As  all  the  amounts  will  not  be  paid  promptly,  it  is  better  to 
make  the  monthly  payments  $2  and  $1.70  respectively. 


io6 


MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 


Serial  Plan 

The  serial  plan  is  virtually  a  succession  of  terminating  plans 
combined  in  one  association.  A  series  is  started  at  regular  inter- 
vals of  three,  six  or  twelve  months,  with  whatever  number  of 
shares  may  be  subscribed.  In  some  states  the  par  of  a  share 
is  $200  and  the  monthly  payment  per  share  is  $1 ;  but  in  most 
of  the  states  the  par  is  $100  and  the  payment  is  50  cents 
per  month,  sometimes  25  cents  per  week.  After  a  new  series 
is  begun  no  more  shares  can  be  issued  in  any  previous  series, 
although  the  holder  of  stock  in  any  series  can  sell  it  or  otherwise 
transfer  it. 

A  serial  association  is  a  partnership  with  limited  liability, 
the  subscribers  to  the  different  series  being  the  partners.  The 
profits  are  divided  among  the  series  in  proportion  to  their 
capitals  which  are  the  amounts  paid  in  plus  the  accumulated 
profits  to  date.  A  simple  illustration  will  make  the  process 
clear. 

Suppose  that  an  association  is  started  on  January  i,  191 7,  and 
that  on  September  30,  1918,  the  shares  outstanding  are  as  fol- 
lows: 


Series 

Date  of 

Issue 

nu.mber  of 
Shares 

Paid  per 
Share 

Profit  per 
Share 

V\\LUE  PER 

Share 

I 

2 

3 

January,  1917 

April, 

July, 

October, 

January,  1018 

April, 

July, 

800 
600 
700 
500 
400 
600 
500 

Jio.so 
9.00 
7. so 
6.00 
450 
3  00 
1.50 

$2 

I 
I 

29 

71 

25 

88 
56 
27 
02 

$12.79 

10.71 

8.7S 

6.88 

s 

6 

7 

5.06 
3-27 

1-52 

During  the  cjuarter  beginning  October  i,  1918,  the  eighth 
series  of  700  shares  is  sold.  The  net  profits  for  the  quarter  from 
interest,  fines,  etc.,  less  the  secretary's  salary  and  other  expenses, 
are  $489.92.    The  undivided  profits  on  September  30,  1918,  are 


BUILDING  AND  LOAN  ASSOCIATIONS  107 

$4,571.28.     There  are  two  plans  by  which  the  profits  may  be 
divided,  the  partnership  plan  and  the  Dexter  rule. 

Distribution  by  Partnership  Plan 

In  the  so-called  partnership  plan  all  previous  profits  are  ig- 
nored and  the  capital  of  each  series  is  taken  as  the  money  actually 
paid  in  multiplied  by  the  equated  number  of  months  during  which 
it  has  been  left  in  the  business.  As  the  dues  are  payable  on  the 
first  of  each  month,  the  equated  time  is  found  by  adding  i  to  the 
number  of  months  the  series  has  run  and  dividing  the  sum  by  2. 
Thus,  on  December  31,  1918,  the  first  series  has  run  24  months. 
Adding  i  and  dividing  by  2  gives  12^^  months  as  the  equated 
time.  As  there  are  800  shares  in  that  series  and  as  each  share  has 
paid  in  $12  the  capital  is  $12  multiplied  by  800  or  $9,600  and  it  has 
been  in  an  equated  time  of  12^  months.  Multiplying  $9,600  by 
12^  gives  $120,000  as  the  equated  capital  of  Series  i.  Pursuing 
the  same  method  the  equated  capitals  of  the  several  series  are 
calculated  as  follows: 

Series 

I $12.00  X  i2>^  mos.  X  800  shares  $120,000 

2 10.50  X  II  "X  600  "  69,300 

3 9.00  X     9/2  "     X  700  "  59,850 

4 7-5°  X  8  "X  500  "  30,000 

5 6.00  X  6>2  "     X  400  "  15,600 

6 4-5°  X  5  "     X  600  "  13,500 

7 3-00  X  3>2  "     X  500  "  5,250 

8 1-50  X  2  "X  700  "  2,100 


Total  of  equated  capitals  $315,600 


The  total  profits  to  date,  amounting  to  $5,061.20,  are  divided 
among  the  series  in  the  proportion  that  each  one's  equated  capital 
bears  to  the  total  equated  capital.  Reducing  to  the  least  com- 
mon denominator  of  2,104,  we  have  the  following  division  of  the 
profits  among  the  several  series. 


io8 


MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 


Series 


I . 

2. 

3- 

soo/ . 

...   462/ 
• • ■    399/ 

4- 

200/ 

5- 
6. 

7- 
8. 

■ ■ ■    104/ 
90/ 
35/ 

. . .    14/ 

800/2104  of  $5,061.20  equals  $1,924.40 


,924.40 

$2.41 

Der  share 

,111-34 

1.85 

959-79 

1-37 

481.10 

.96 

250.17 

.62 

216.50 

-36 

84.19 

-17 

33-68 

•OS 

2,104 


;,o6i.i7 


Of  course  it  is  impossible  to  balance  exactly  the  total  profits 
with  the  sum  of  the  profits  assigned  to  each  series. 

Distribution  by  Dexter's  Rule 

The  first  step  in  applying  the  Dexter  rule  is  to  calculate  the 
capitals  of  the  series.  The  capital  of  each  series  on  September  30 
is  the  value  per  share  as  shown  in  the  table  on  page  106  [of  this 
chapter]  multipHed  by  the  number  of  shares  outstanding.  To  find 
the  earning  capital  on  December  31,  it  is  necessary  to  add  the 
contribution  of  50  cents  per  share  for  each  of  the  three  months 
of  the  current  quarter.  This  is  an  average  of  $1  per  share,  cal- 
culated thus : 

50  cents  paid  October       i  was  in  3  months  or  $1.50  for  i  month 

50     "        "      November  i     "     "2       "         "     i.oo    "   i 

50     "        "      December  i     "      "  i  month     "       .50    "   i        " 


Total $3.00 


This  is  an  average  of  $1  for  three  months.  One  dollar  must, 
therefore,  be  added  to  the  value  per  share  shown  in  the  first  table. 
Bearing  this  in  mind,  the  earning  capital  of  the  different  series  is 
found  as  follows: 


BUILDING  AND  LOAN  ASSOCIATIONS 


109 


Series 

I 800  shares  at  $13.79  P^r  share 

"  II. 71 


600 
700 
500 
400 
600 
500 
700 


9-75 
7.88 
6.06 
4.27 
2.52 
1. 00 


)II,032 

7,026 

6,825 
3,940 
2,424 
2,562 

1,260 
700 


Total  capital $35,769 


This  rule  leaves  undisturbed  the  previous  distributions  of 
profits  and  adds  to  them  the  profits  of  the  current  quarter  on  the 
basis  of  the  present  earning  capitals.  The  current  profits  of 
$489.92  are  1.37  per  cent  of  the  capital  of  $35,769.  Hence  the 
profit  per  share  for  each  series  is  calculated  as  follows : 


Series 

Capital 

Current 
Profit 

Previous 
Profit 

Total 
Profit 

Shares 

Profit  per 
Share 

$11,032 
7,026 
6,825 
3.940 
2,424 
2,562 
1,260 
700 

$151.14 
96.26 
93-50 
53-98 
33-21 
35-10 
17.26 

9-59 

$1,832 
1,026 
875 
440 
224 
162 
10 

$1,983.14 
1,122.26 
968.50 
493-98 
257.21 
197-10 
27.26 

9-59 

800 
600 
700 
500 
400 
600 
500 
700 

$2.48 

1.87 

1.38 

-99 

.64 

6 

■  33 

•  05 

8      

.01 

$35,769 

$490.04 

$4,569 

$5,059-04 

A  comparison  of  the  profits  per  share  shows  that  the  partner- 
ship plan  gives  larger  profits  to  the  later  at  the  expense  of  the 
earlier  series.  This  is  a  great  injustice,  as  the  earlier  series  are 
made  to  share  the  profits  they  have  accumulated,  with  the  later 
series,  which  had  no  part  in  producing  those  profits.  This  is  not 
done  in  an  ordinary  commercial  business  when  new  partners  are 
admitted.  Dexter's  rule  is  the  true  partnership  plan  and  is  the  one 
usually  adopted. 


no         MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 

Withdrawal  of  Shares 

In  case  any  shares  are  withdrawn  during  the  quarter  the 
holders  are  given  the  amount  they  have  paid  in  and  interest  for 
the  equated  time  at  3  or  4  %.  Thus,  if  fifty  shares  in  the 
third  series  are  withdrawn  on  December  i,  191 8,  the  amount  paid 
in  is  $8.50  per  share  or  $425,  the  interest  at  3%  for  the  equated 
time  of  nine  months  is  $9.56,  and  the  total  amount  paid  is 
$434.56.  The  book  value  of  this  stock  is  $9.75  per  share  or 
$487.50.  There  is  a  book  profit  of  $5 2. 94  on  the  transaction.  No 
entry  is  made  for  this  profit,  but  it  is  added  to  the  cash  profits 
of  $489.92,  which  makes  the  divisible  profits  for  the  quarter 
$542.86. 

Sources  of  Income 

The  profits  of  building  and  loan  associations  arise  from  inter- 
est paid  by  the  borrowers  and  also  from  fees,  fines  and  premiums. 

The  fees  are  membership  fees,  usually  25  cents  per  share,  paid 
when  the  stock  is  originally  subscribed.  Sometimes  these  fees 
are  capitalized  in  the  series  in  which  they  are  paid,  making  the 
value  per  share  25  cents  more.  Otherwise  they  are  treated  as  a 
general  profit,  as  in  our  example.  In  some  associations  a  transfer 
fee  is  paid  when  stock  changes  hands. 

The  fines  are  levied  on  stockholders  who  are  delinquent 
in  their  monthly  payments,  usually  5  cents  per  share  on 
stock  of  non-borrowers  (investment  stock),  and  10  cents  per 
share  on  stock  of  borrowers,  for  each  month  the  delinquency 
continues. 

Premiums 

The  premiums  are  either  payable  monthly  or  are  deducted 
from  the  loan  when  made.  They  arise  from  the  bidding  for  loans 
by  those  desiring  to  borrow.  When  the  association  is  in  posses- 
sion of  loanable  funds,  bids  are  invited.  The  person  offering  to 
pay  the  highest  premium  for  the  money  receives  the  loan,  if  his 


BUILDING  AND  LOAN  ASSOCIATIONS  III 

real  estate  security  is  found  to  be  adequate.  On  the  cash  pre- 
mium plan,  1%  of  the  premium  is  paid  monthly  in  cash.  On 
the  deducted  premium  plan,  the  whole  premium  is  deducted  from 
the  loan  in  advance  and  only  the  net  amount  is  paid. 

If  the  loanable  funds  are  $2,000  and  the  cash  premium  bid  is 
25%,  the  borrower  makes  his  note  for  $2,000  and  his  monthly 
payments  are:  Dues  $10,  interest  at  6%  $10,  and  premium 
amounting  to  1%  of  $500  premium  or  $5,  making  a  total  cash 
payment  of  $25. 

For  the  same  amount  of  money  on  the  deducted  premium 
plan,  the  bid  may  be  for  $2,500  at  a  premium  of  20%.  The 
borrower  receives  $2,000  cash,  but  his  note  is  made  for  $2,500  and 
his  monthly  payments  are:  Dues  $12.50  and  interest  $12.50,  a 
total  cash  payment  of  $25.  The  premium  of  $500  is  credited  to 
unearned  premiums,  and  at  the  end  of  each  quarter  profit  and  loss 
is  credited  and  unearned  premiums  charged  with  $15. 

Each  borrower  is  obliged  to  subscribe  for  stock  the  par  value 
bf  which  is  equal  to  the  face  of  his  loan.  When  the  dues  paid  on 
this  stock  and  the  profits  credited  to  it  cause  it  to  be  worth  par, 
it  is  paid  off  by  cancelling  the  loan.  No  payments  are  ever 
applied  to  the  loan,  which  remains  in  full  force  until  cancelled  by 
the  matured  stock. 

It  will  be  noted  that  in  both  the  serial  plans  the  treatment  is 
entirely  on  the  basis  of  the  several  series  being  fully  paid  to  date. 
If  stock  in  any  series  is  delinquent,  it  still  receives  its  share  of  the 
profit,  the  only  offset  being  the  small  fine,  which  in  the  older 
series  is  much  less  than  the  earnings  for  the  quarter. 

Individual  Plan 

To  meet  this  objection  to  the  serial  plan,  some  associations 
have  adopted  the  individual  plan,  which  really  makes  each  stock- 
holder a  series  by  himself.  The  capital  is  then  the  actual  amount 
paid  in  plus  accumulated  profits.  The  procedure  as  to  earning 
capital  and  profits  is  otherwise  the  same  as  in  the  serial  plan. 


112         MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 

Dayton,  or  Ohio  Plan 

One  great  objection  to  the  plans  hitherto  considered,  or  to 
any  modification  of  them,  is  that  they  place  the  borrowers  in 
an  uncertain  position  with  regard  to  the  time  at  which  their  loans 
will  be  cancelled,  and  consequently  with  regard  to  the  amount  of 
principal  and  interest  they  will  be  called  upon  to  pay.  Their 
loans  are  not  cancelled  until  their  stock  matures.  If  the  associ- 
ation proves  to  be  very  prosperous,  the  profits  will  be  large  and  the 
stock  will  mature  in  a  comparatively  short  time.  On  the  other 
hand,  in  a  poorly  managed  association,  it  may  take  a  long  time  to 
mature  the  stock,  and  in  the  meantime  the  borrowers  will  have  to 
continue  the  payment  of  their  instalments  and  interest,  making 
their  loans  much  more  expensive  than  they  have  anticipated. 

To  meet  this  objection  various  forms  of  the  "Dayton  Plan" 
have  been  adopted,  all  of  which  agree  in  this — that  the  borrower 
is  given  a  definite  contract.  His  loan  is  to  be  repaid  in  a  fixed 
number  of  months  by  the  regular  payment  of  a  definite  amount, 
which  will  cover  both  principal  and  interest.  The  calculation  of 
the  amount  to  be  paid  is  based  on  the  same  principle  as  the  one 
already  shown  for  liquidating  a  church  debt.  The  amount 
applicable  to  the  principal  is  credited  to  the  loan  each  month  by 
the  association,  as  the  borrower  does  not  have  to  own  any  stock. 
The  loanable  funds  are  obtained  from  the  instalments  paid  by 
investing  stockholders,  with  whom  the  accounts  are  kept  on  the 
individual  plan. 

This  system  is  really  that  of  a  co-operative  savings  bank,  and 
is  the  fairest  method  of  all.  When  properly  managed,  it  is  very 
successful. 

The  two  following  problems  illustrate  principles  dealt  with  in 
this  chapter. 

Problem  i 

A  building  and  loan  association  published  the  following  statement  on 
December  31,  iqi8: 


BUILDING  AND  LOAN  ASSOCIATIONS 

Age  and  Condition  of  Shares 


113 


Series 

Date  of 

Issue 

Number  of 
Shares 

Paid  per 

Share 

Profit  per 
Share 

Value  per 
Share 

I 

January,  191 7 
April, 
July, 

October,      " 
January,  1918 
April, 
July. 
October,      " 

800 
600 
700 
Soo 
400 
600 
500 
600 

$12.00 
10.50 
9.00 
7.50 
6.00 
4-50 
3.00 
I. so 

$2.56 

1.97 

ISI 

1. 14 

.82 

•  53 

.26 

.02 

$14.56 
12.47 
10.51 

3 

4--- 

s 

8.64 
6.82 

6 

3.26 

8 

The  balance  sheet  showed  that  the  profits  to  date  were  $5,647.82. 
On  March  31,  1919,  the  net  profits  from  interest,  premiums,  fines  and  fees, 
less  salary  and  expenses,  were  found  to  be  $483.16.  There  had  been  with- 
drawn during  the  quarter  50  shares  of  the  4th  series  on  which  $10.75  ^^- 
terest  had  been  paid,  not  included  in  the  expenses  previously  mentioned. 
The  Qth  series  was  opened  and  400  shares  were  subscribed  and  paid  for  the 
quarter. 

Distribute  the  profits  March  31,  1919,  on  the  partnership  plan. 

Solution: 

First  find  the  equated  time;  which  is  ascertained  by  adding  i  to  the 
number  of  months  in  the  age  of  each  series  and  dividing  by  2.  Next  multi- 
ply the  equated  time  by  the  amount  paid  and  then  by  the  number  of 
shares  outstanding  in  each  series,  as  follows: 


Series 

I . . 

••     27 

months  old. 

equated 

14      mos.  X  3 

13.50  X  800  si 

1.    $151,200 

2. . 

..     24 

I2>2 

'    X 

12.00  X  600 

90,000 

3-- 

21 

II 

'    X 

10.50  X  700 

80,850 

4-- 

..     18 

9>4 

'    X 

9.00  X  450 

38,475 

S-- 

•      15 

8 

'    X 

7.50  X  400 

24,000 

6.. 

12 

6>^ 

'    X 

6.00  X  600 

23,400 

7- 

9 

5 

"    X 

4.50  X  500 

11,250 

8.. 

6 

s'A 

"    X 

3.00  X  600 

6,300 

9.. 

••       3 

2 

'     X 

1.50  X  400 

1,200 

$426,675 

114 


MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 


The  profits  to  be  divided  are  found  as  follows: 

Undivided  profits,  December  31,  1918 

Profits  of  quarter,  net 


Less  interest  paid  on  withdrawal 

Profits  to  be  divided 

The  profits  are  divided  in  the  following  manner: 
Series 


;,647. 82 

483-16 

),i30.98 

IO-75 

3,120.23 


2oi6/5689*of|6,i2o.23equals$2,i68.8i,orpershare       $ 


1200 
1078 

513 
320 
312 

150 

84 

16 

5,689 


,290.96  ' 

,159-71 ' 

551-89' 

344.26 ' 

335-65 ' 

u     u 

161.37 ' 

u     u 

90.37 ' 

17.22 ' 

2.71 

2-15 

1.66 

1.23 

.86 

•56 

•32 

•15 
.04 


),I20.24 


*i5i, 200/426,675  =  2016/5689 

Problem  2 

Using  the  same  figures  as  in  the  previous  problem,  distribute  the  profits 
in  accordance  with  Dexter's  rule. 

Solution: 

First  find  the  capitals  of  the  series  by  multiplying  the  value  per  share 
plus$i  by  the  number  of  shares  outstanding,  as  follows: 
Series 

800  shares  by  $15.56  equals  $12,448 


600 
700 

450 
400 
600 
500 
600 
400 


13-47 

8,082 

II. 51 

8,057 

9-64 

4,338 

7.82 

3,128 

6.03 

3,618 

4.26 

2,130 

2.52  ' 

1. 512 

1. 00 

*     400 

$43,713 

BUILDING  AND  LOAN  ASSOCIATIONS 


115 


The  profits  for  the  quarter  are $483.16 

Plus  profit  on  withdrawn  shares 46.25 

Total  to  be  divided $529.41 


The  withdrawal  profits  are  ascertained  thus: 

Book  profit  for  4th  series  is  $1.14  per  share,  and  for  50  shares. 
Actually  paid 

Retained  profit  on  withdrawal 


>S7-oc 
10.75 


'■25 


The  divisible  profit  of  $529.41  is  a  trifle  more  than  1.21%  of  the 
capital  of  $43,713.    Calculated  for  each  series  it  is  as  follows: 


Series 

Shares 

Old  Profit 
PER  Share 

Old  Profit 
OF  Series 

New  Profit 
OF  Series 

Total    Profit 
per  Share 

New  Profit 
PER  Share 

I 

800 

J2.S6 

$2,048.00 

J150.62 

$2,198.62 

$2.75 

2 

600 

1-97 

1,182.00 

97 

79 

1,279.79 

2.13 

3 

700 

l-Sl 

1,057-00 

97 

49 

1. 154-49 

1-65 

4 

450 

1. 14 

51300 

52 

49 

565-49 

1.26 

S 

400 

.82 

328.00 

37 

85 

365-85 

-91 

6 

600 

.53 

318.00 

43 

78 

361.78 

.60 

7 

500 

.26 

130.00 

25 

77 

155-77 

•  31 

8 

600 

.02 

12.00 

18 

30 

30-30 

■OS 

9 

400 

0 

4 

84 

4.84 

.02 

$5. 588. 00 

S528 

93 

$6,116.93 

The  last  series  is  usually  given  2  cents  as  i  cent  is  so  very  small. 
With  the  profit  per  share  known  the  statement  of  age  and  condition 
of  shares  can  now  be  made  up  if  asked  for. 


CHAPTER   XIII 

GOOD-WILL   AND    CONSOLIDATIONS 

Purchasing  a  Business  with  Stock 

When  two  or  more  concerns  are  merged,  or  when  a  business  is 
taken  over  by  a  holding  company,  or  when  a  business  is  bought 
by  a  corporation  already  in  existence,  the  payment  for  the  busi- 
ness taken  over  is  frequently  made  in  the  stock  of  the  corporation 
acquiring  the  business. 

If  the  amount  of  stock  to  be  given  for  the  business  is  to  be 
determined  on  an  equitable  basis,  two  elements  in  the  business 
must  be  taken  into  consideration.  These  are  the  fair  value  of  the 
net  assets  of  the  business  as  a  going  concern  and  the  earning  ca- 
pacity of  the  business. 

The  fair  value  of  the  net  assets  may  be  reached  by  mutual 
agreement  between  the  parties,  or  by  an  appraisal  of  the  fixed 
assets  by  an  appraisal  company,  and  an  estimate  of  the  value  of 
the  active  assets  by  an  accountant. 

When  the  value  is  determined,  stock  of  a  corresponding 
amount  in  face  value  is  allotted  as  the  price  paid  for  the  net  assets. 

Allocation  of  Net  Earnings 

To  ascertain  the  earning  power  for  which  further  stock  should 
be  issued,  it  is  first  necessary  to  establish  a  normal,  or  standard 
rate  of  earnings.  This  can  be  done  only  by  agreement  between 
the  parties  concerned.  When  the  rate  is  agreed  upon,  it  is  ap- 
plied to  the  stock  already  allotted  for  net  assets.  The  result  is 
the  amount  of  net  earnings  to  be  devoted  annually  as  dividends 
on  the  stock  given  in  payment  of  the  net  assets. 

The  net  earnings  remaining  after  deducting  dividends  at  the 
agreed  rate  on  the  stock  issued  for  net  assets,  is  the  basis  for 

Ii6 


GOOD-WILL  AND  CONSOLIDATIONS  II7 

calculating  the  amount  of  stock  to  be  allotted  for  the  excess 
earning  capacity  of  the  business  acquired. 

The  theory  on  which  stock  is  allotted  for  excess  earnings  is 
based  on  the  fact  that  one  of  the  principal  reasons  for  investing 
in  a  stock  is  that  it  will  produce  an  income  that  is  considered 
adequate  return  on  the  money  invested.  What  the  rate  of  return 
should  be  is  determined  by  the  conditions  of  the  enterprise.  An 
investment  in  a  business  whose  conditions  are  stable  and  which 
may  be  depended  upon  to  maintain  a  steady  earning  power,  or 
even  to  increase  it,  yields  a  much  lower  rate  of  return  than  one  in 
a  speculative  and  unreliable  business. 

Good-Will 

The  name  given  to  the  valuecreatedby  the  excess  earnings  of  a 
business  is  "good- will."  Consideration  of  good- will  is  involved  in 
the  discussion  of  a  stock  distribution  on  the  basis  of  earnings. 
The  stock  allotment  is  partly  based  on  the  value  of  the  net  assets 
acquired  and  partly  on  the  good-will,  or  excess  earning  power. 

Since  good-will  is  the  measure  of  earning  capacity,  it  follows 
that  only  an  established  business  can  possess  it.  It  resides 
usually  in  the  reputation  the  business  has  built  up  in  consequence 
of  the  excellence  of  its  product,  the  practice  of  fair  dealing  or  any 
other  characteristics  of  its  management  giving  it  an  advantage 
over  its  competitors.  The  existence  of  good-will  cannot  be 
proved  except  by  a  more  or  less  prolonged  experience.  There- 
fore, when  a  new  business  is  started  it  is  false  accounting  to  issue 
any  stock  for  good-will,  because  its  promoters  expect  it  to  earn 
more  than  the  normal  rate  of  profit.  A  new  company,  however, 
which  has  taken  over  an  old  business  with  a  developed  good-will 
may  carry  good-will  among  its  assets  and  issue  stock  therefor. 

Appraising  Good- Will — Years'  Purchase  Method 

In  allotting  stock  the  good-will  may  be  valued  in  several 
different  ways.     It  may  be  expressed  as  a  given  number  of  years' 


Il8         MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 

purchase  of  the  total  profits.  This  means  that  the  purchaser  is 
wilHng  to  forego  profits  from  the  business  for  the  time  agreed 
upon.  Each  condition  of  the  terms  must  be  agreed  upon  between 
the  buyer  and  seller.  Thus,  if  it  is  agreed  that  the  basis  shall  be 
two  and  one  half  years'  purchase  of  the  average  profits  for  the 
last  ten  years,  it  will  be  necessary  to  ascertain  the  total  profits 
for  ten  years.  One  tenth  of  this  sum  will  be  the  average  yearly 
profit,  and  two  and  one  half  times  the  average  profit  will  be  two 
and  one  half  years'  purchase. 

If  the  total  profit  for  lo  years  is  $256,232.00 

the  average  annual  profit  is  $  25,623.20 

and  23^  times  the  average  is  $  64,058.00 

which  is  the  value  placed  upon  the  good-will  on  the  basis  of  two 
and  one  half  years'  purchase. 

It  is  better  to  ascertain  the  average  of  a  number  of  years  than 
to  take  the  figures  for  the  last  two  and  one  half  years,  as  the 
latter  may  not  be  normal. 

This  method  is  not  as  logical  as  allowing  a  certain  number  of 
years'  purchase  of  the  profits  in  excess  of  a  rate  agreed  upon  as 
normal.  The  latter  calculation  is  the  same  as  that  given  above, 
except  that  the  normal  profits  are  first  deducted.  Thus,  if  the 
invested  capital  is  $200,000  and  the  normal  rate  agreed  upon  is 
10%,  the  value  of  the  good-will  is  figured  as  follows: 

Average  profits  (as  shown  above) ....     $25,623.20 
Normal  profit 20,000.00 

Excess  profit $  5,623.20 

8-years'  purchase $44,985.60 

Capitalizing  Gross  Income 

A  third  method  is  to  capitalize  the  gross  income.  If  the 
capital  is  known  and  the  per  cent  of  income  is  also  known,  the 
income  is  found  by  multiplying  the  capital  by  the  rate  per 
cent.     Thus,  if  the  capital  is  $200,000  and  the  rate  of  profit  is 


GOOD-WILL  AND  CONSOLIDATIONS  119 

15%,  the  income  is  $30,000.  On  the  other  hand,  if  it  is  desired 
to  ascertain  what  capital  will  produce  $30,000  income  if  the  rate 
per  cent  is  15,  it  is  necessary  to  divide  the  income  by  the  per  cent 
expressed  as  a  decimal,  thus, 

$30,000  divided  by  .15  equals  $200,000. 

Therefore,  to  capitalize  a  given  income  at  a  given  rate  per  cent, 
the  income  is  divided  by  the  decimal  expressing  the  rate. 

This  method,  like  the  first,  is  objectionable  because  it  does  not 
take  into  consideration  the  fact  that  part  of  the  income  must  be 
used  to  furnish  a  return  on  the  stock  already  allotted  for  net 
assets.  That  amount  of  the  income  is  in  fact  duplicated,  and  the 
duplication  reduces  the  amount  of  the  return  to  the  concern  with 
the  largest  rate  of  profit  and  increases  the  return  to  the  concern 
with  the  smallest  rate. 

Suppose  the  following  companies  wish  to  combine  on  the  basis 
of  net  assets  and  good-will  representing  gross  income  capitalized 
at  15%. 

ABC 

Net  assets  and  capital $200,000     $300,000     $500,000 

Average  annual  income 40,000         45,000         65,000 

Rate  of  profit 20%  15^^  13% 

If  it  is  proposed  to  form  a  new  company  with  a  capital  of 
$2,000,000,  the  distribution  of  the  stock  is  as  follows: 

For  net  assets $200,000     $300,000     $500,000 

For  income  capitalized  at  15%. .       266,667       300,000       43i^3,.^3 

Total $466,667     $600,000     $933,333 

The  total  income  being  $150,000,  the  rate  of  profit  is  7>^  %, 
which  will  have  to  be  distributed  as  follows: 

A  receives  y}4  %  of $466,667       $35,000 

B      "  "      "    " 600,000        45,000 

C      "  "      "    " 933^333         70.000 

Total $2,000,000    $150,000 


120         MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 

A  would  thus  receive  $5,000  less  per  year  than  when  operating 
alone  and  will  certainly  object  to  the  plan.  B  would  not  be 
affected  because  he  is  already  receiving  15%,  while  C  would  gain 
the  $5,000  lost  by  A. 

Therefore,  a  fourth  plan  should  be  adopted  by  which  the 
duplication  of  profits  will  be  avoided.  After  allowing  for  a  fixed 
rate  of  return  on  the  capital  issued  for  net  assets  to  each  of  the 
merging  concerns,  the  amount  of  profits  necessary  to  provide  the 
dividend  at  that  rate  is  deducted  from  the  total  profits  previously 
earned  by  the  concern  and  the  capitalization  of  the  remaining 
profits  is  made  by  employing  the  same  rate.  This  amounts  to 
fixing  the  rate  desired  on  the  new  stock  and  issuing  stock  to  each 
concern  sufficient  to  earn  the  profits  previously  enjoyed.  If  8% 
is  fixed  upon  as  the  rate  desired,  the  first  allotment  of  stock 
for  net  assets  is  the  same  as  before — A  $200,000,  B  $300,000,  and 
C  $500,000.  The  remaining  profits  are  then  calculated  as 
follows : 


A 

B 

C 

Total 

$  40,000 
16,000 

$  45,000 
24,000 

$  65,000 
40,000 

$ 

150,000 

80,000 

$  24,000 

$  21,000 

$  25,000 

$ 

70,000 

Capitalizing  at  8% 

$300,000 
200,000 

$262,500 
300,000 

$312,500 
500,000 

$ 
] 

875,000 

,000,000 

$500,000 

$562,500 

$812,500 

$ 

,875,000 

Income  at  8% 

$  40,000 

$  45,000 

$  65,000 

$ 

150,000 

In  this  way  each  concern  will  receive  the  same  earnings  after  the 
merger  as  before,  and  none  will  suffer. 


Issue  of  Two  Classes  of  Stock 

We  have  thus  far  considered  the  issue  of  only  one  kind  of  stock. 
This  necessitates  the  use  of  only  one  rate  of  return  on  the  stock 


GOOD-WILL  AND  CONSOLIDATIONS  121 

issued.  A  more  usual  procedure  when  a  combination  of  this  kind 
is  effected  is  the  issuance  of  two  classes  of  stock,  preferred  and 
common.  It  is  an  almost  universal  practice  to  issue  preferred 
stock  for  the  net  assets  and  common  stock  for  the  good-will,  or 
excess  earnings. 

The  objection  to  calculating  the  stock  given  for  good-will  on 
the  basis  of  the  total  earnings  is  the  same  as  in  the  third  plan. 
If  the  preferred  stock  is  issued  at  6%  and  the  good-will  is  capital- 
ized on  the  total  profits  at  20%,  the  common  stock  issued  for 
good-will  is  as  follows: 

A $200,000 

B 225,000 

C 3  25^000 

Total $750,000 


As  it  takes  $60,000  to  pay  the  preferred  dividend,  there  will  be 
left  $90,000  for  the  common  stock,  which  will  allow  a  12%  divi- 
dend.    The  profits  will  therefore  be  divided  as  follows: 

A $200,000  preferred  at    6%     $12,000 

200,000  common   at  12%       24,000       $36,000 

B 300,000  preferred  at    6% 

225,000  common  at  12% 

C 500,000  preferred  at    6% 

325,000  common   at  12% 

Total 


$18,000 

27,000 

45,000 

$30,000 

30,000 

6g,ooo 

$: 

[50,000 

In  this  way  A  makes  a  loss  and  C  a  gain  of  $4,000. 

The  proper  method  is  to  issue  the  common  by  capitalization 
of  the  excess  profits  after  the  preferred  dividends  have  been 
provided  for,  thus: 

A 

Original  profits $40,000 

Preferred  dividends  6% 12,000 

Excess  remaining $28,000       $27,000       $35,000 

Capitalized  at  20% $140,000    $135,000    $175,000 


B 

C 

$45,000 

$65,000 

18,000 

30,000 

122         MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 

It  is  evident  that  dividends  of  6%  on  the  preferred  and  20% 
on  the  common  stock  will  give  the  same  return  as  formerly. 

After  the  preferred  dividends  are  deducted  it  does  not  make 
any  difference  what  basis  is  adopted  for  capitalizing,  because 
whatever  rate  is  used  the  relative  proportion  remains  the  same. 
Thus,  if  the  excess  is  capitalized  at  four  years'  purchase  the  result- 
ant common  stock  is  as  follows:  A  $112,000,  B  $108,000,  and  C 
$140,000,  a  total  of  $360,000,  on  which  the  $90,000  remaining 
profits  will  provide  a  dividend  rate  of  25%,  and  the  distribution 
will  be  thus: 

A  for  preferred $12,000 

"   common,  25%  of  $112,000 28,000     $40,000 

B    "  preferred $18,000 

"  common,  25%  of  $108,000 27,000       45,000 

C    "  preferred $30,000 

"  common,  25%  of  $140,000 35, 000       65,000 

The  practice  of  starting  a  new  business  with  a  capital  stock  of 
a  par  value  greater  than  the  total  of  the  net  assets  is  often  in- 
dulged in.  In  such  a  case  a  debit  must  be  made  to  some  account, 
in  order  to  balance  the  books.  The  account  debited  is  often 
called  good-will.  This  is  altogether  wrong,  since,  as  we  have 
seen,  good- will  is  a  matter  of  growth  and  cannot  be  possessed  by  a 
business  which  has  had  no  time  to  develop  it. 


CHAPTER  XIV 
FOREIGN   EXCHANGE 

Conversion  of  Foreign  Coinage 

In  order  to  provide  a  basis  for  the  comparative  values  of 
United  States  and  foreign  coinage,  the  Director  of  the  United 
States  Mint  periodically  estimates  the  values  of  foreign  coins, 
which  are  proclaimed  by  the  Secretary  of  the  Treasury.  The 
values  thus  fixed  are  called  the  mint  pars  of  exchange,  and  repre- 
sent the  intrinsic  or  bullion  values  of  foreign  coins  in  terms  of 
United  States  money.  A  table  of  values  recently  proclaimed 
appears  in  the  Appendix. 

To  reduce  a  value  in  foreign  coinage  to  United  States  money, 
multiply  the  value  of  the  coin  in  United  States  money  by  the 
number  of  coins. 

Illustration 

What  will  a  draft  for  £410 'i9/'6  cost  in  dollars  at  the  mint  par  of 
exchange? 

Solution:  There  are  two  ways  to  calculate  this: 

1.  Reduce  shillings  and  pence  to  decimals  of  the  pound,  thus: 

£410. 

ig  shillings  are  19/20  of  a  pound .95 

6  pence  are  6/240  or  1/40  of  a  pound .025 

£410.975 
£410.975  multiplied  by  $4.8665  is  $2,000.01. 

2.  Reduce  pence,  shillings  and  pounds  to  value  in  dollars,  thus: 

6  pence  are  1/40  of  $4.8665 $  .12 

19  shillings  are  19/20  of  $4.8665 4.62 

410  pounds  at  $4.8665 1,995.27 

£410/19/6  at  $4.8665 $2,000.01 

123 


124         MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 

Reverse  Conversion 

To  reduce  a  value  in  United  States  coinage  to  a  foreign  money 
value,  divide  the  value  in  United  States  money  by  the  value  of 
the  foreign  unit. 

Illustration 

What  is  the  value  in  pounds  of  $2,000  at  the  mint  par  of  exchange? 

Solution:  There  are  again  two  ways  to  calculate  this,  which  are  as 
follows : 

1.  2,000 -^  4.8665  =  410.973,  number  of  pounds 

.973  of  20  (shillings  in  the  pound)  =  19.46 
.46  of  12  (pence  in  the  shilling)  =  5.52 

Hence  $2,000  =  £410/19/6. 

2.  To  find  the  number  of  pounds: 

4.8665  1  2,000.00  I  4io£ 


1,946.60 

53-400 

48.665 

Remainder 

4-735° 

Reduce  to  shillings 

20 

48665    947000  1  19  s 

48665 

460350 

437985 

Remainder 

22365 

Reduce  to  pence 

12 

48665I 268380  1  5.52  d 

243325 

250550 

243325 

72250 

Result  £  4 1 0/19/6. 
In  this  particular  case  there  is  a  much  shorter  method,  which  is  as 
follows : 


FOREIGN  EXCHANGE  1 25 

£410/19/6  is  the  same  as  £411  minus  6d,  or  £411  minus  1/40  of  £1. 
We  can,  therefore,  obtain  the  result  by  performing  the  following  sub- 
traction : 

£4ii/oo'o  X  $4.8665 $2,000.13 

—  6  or  1/40  of  $4.8665 —.12 

£410/19/6 $2,000.01 

Current  rates  of  exchange  vary  from  the  unit  par  rate,  because 
the  supply  and  demand  for  foreign  drafts  are  afifected  by  the 
balance  of  trade  between  countries.  Transactions  involving 
foreign  exchange  are  made  at  current  instead  of  mint  par  rates. 

Exchange  is  quoted  at  the  current  value  in  cents  and  con- 
versions are  made  as  illustrated  in  the  conversions  at  mint  rates. 

Illustration 

What  is  the  cost  of  a  300  franc  draft  on  Paris,  purchased  at  8.49? 

Solution:    $.0849  X  300  =  $25.47 

A  United  States  merchant  sends  $5,000  to  Paris  when  the  rate  is  8.49. 
What  is  the  value  in  francs? 

Solution:  5,000 -^  .0849=  58,892.8,  the  number  of  francs. 

Dealing  in  Foreign  Exchange 

There  are  two  methods  of  recording  transactions  in  foreign 
exchange.  When  the  first  method  is  used,  the  charges  and  credits 
are  entered  in  two  values,  domestic  and  foreign,  the  conversion 
of  each  item  being  made  at  the  rate  applying  to  the  transaction 
When  the  account  is  closed  the  balance  in  foreign  money  is 
valued  by  converting  either  at  cost  or  current  rate,  and  the  profit 
or  loss  on  exchange  is  ascertained. 

The  other  method  attempts  to  ascertain  the  profit  or  loss  on 
each  item,  by  computing  the  difference  between  the  conversion 
value  at  par  and  current  rates.  Both  methods  are  illustrated 
below : 


126         MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 


Illustration 

A  banking  concern  deals  in  foreign  exchange  and  the  following  are  the 
transactions  with  a  London  correspondent  for  one  month: 

Debits 

Sept.     1     Remittance,  30-day  bill  £400  at   $4.86 

10  "  sight  bill      £100/10       "     4-87 

"15  "  "       "         £200/0/6     "     4-86K 

Credits 
Sept.     2     Draft,  sight  £300  at  $4.87^ 


12 


£200/12/5  "     4-87 


20     Cable  £100  "     4.88 

1.  Ascertain  the  profit  or  loss  in  the  account  for  the  month. 

2.  State  the  balance  of  the  account  at  the  end  of  the  month  in  foreign 
and  domestic  currency,  the  current  rate  of  sterling  exchange  for 
cable  transfers  being  $4.89. 

Solution  i  :  Although  the  problem  states  that  the  current  rate  of 
exchange  for  cable  transfers  is  $4.89,  it  seems  incorrect  to  inventory  the 
balance  of  the  account  at  this  rate,  since  the  rate  is  not  only  higher  than 
cost  but  higher  than  any  of  the  prices  at  which  sales  have  been  made.  On 
the  theory  that  the  first  items  purchased  are  the  first  sold  we  find  that  the 
first  two  remittances  are  ofTset  by  the  first  two  credits.  Applying  this 
theory,  the  balance  of  the  account  must  have  cost  $4.86^,  the  rate  apply- 
ing to  the  remittance  of  September  15.  Setting  up  the  account  and  valu- 
ing the  balance  at  $4.86^,  we  have  the  following: 

Account  with  London  Correspondent 
Debits 

Foreign  Rate  Domestic 

Sept.    I,  Remittance,  30-day  bill £400/00/0  $4.86  $1,944.00 

"       xo,            "           sight  bill loo/io/o       4.87  489.44 

"       15,            "              "        " 200/00/6       4.86K  973.62 

"       30,  Profit  6.74 

£700/10/6  $3,413.80 


Oct.      I,  Balance — Inventory £99/18/1     $4.86^        $486.28 


FOREIGN  EXCHANGE  127 

Credits 

Sept.    2,  Draft,  sight £300/00/0  $4,871^    $1,462.50 

"       12,       "         " 200/12/5  4.87              977-02 

"       20,  Cable 100/ 00/0  4.88              488.00 

"       30,  Balance 99/18/1  4-86^         486.28 

£700/10/6  $3,413.80 


While  $6.74  is  shown  as  profit  it  must  be  remembered  that  the  rate 
of  4.86^  is  used  because  the  current  rate  for  checks  is  unknown,  that  interest 
has  been  disregarded  in  the  calculation,  and  that  the  profit  as  shown  is 
nominal  only  and  may  be  increased  or  decreased  when  realized  by  a  sale. 

If  the  balance  were  inventoried  at  $4.89,  its  domestic  value  would  be 
$488.53,  and  the  profit  would  be  $8.gg. 

It  would  be  correct  to  value  the  balance  at  the  current  buying  rate  on 
September  30,  if  that  were  known.     This  is  the  usual  practice. 

In  calculating  the  dollar  value  of  the  balance,  the  shortest  way  is  to 
take  £99/18/1  as  I  s  11  d  less  than  £100/00/0,  as  follows: 

I  shilling  is  1/20  of  $4.8675  or $0.24337 

II  pence  is  1/12  less  than  i  shilling,  that  is, $.24337 

—  .02027         .22310 


5.47 


£100  $486.75 

-  i/ii  --47 

£  99/18/1 


Solution  2:  The  above  is  the  usual  method  of  keeping  a  foreign  ex- 
change account  in  this  country.  In  Canada  each  transaction  as  recorded 
in  the  buying  and  selling  registers  is  compared  with  the  par  of  exchange 
and  an  entry  made  debiting  or  crediting  Exchange,  as  the  case  may  be. 
The  par  of  sterling  is  $4.8665  or  $4.86?/^.  The  registers,  in  the  example 
taken,  would  show  the  following: 

Bought 
Date  Time      Sterling       R.ate  Paid  Par  Dr.  Ex.    Cr.  Ex. 

Sept.    1 30  d     £40000/0       $4.86       $1,944.00  51,946.67       S  $2.67 

10 St.  loo/io/o         4.87  489.44  489.10  .34 

IS "  200/00/6  4-8634         973.62  973-45  -'7 

£700/10/6  J3,409-22         $.51  $2.67 


128         MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 

Sold 


Sept.     2  ..  .  . 
12  ...  . 
20..  .   . 
30 

St. 

.  .  .  .        ca. 
....  balance 

£300/00/0 

200/12/5 

loo/oo/o 

99/18/1 

4.87 
4.88 

$1,462.50 

977-02 
488.00 

$1,460.00 
976.35 
486.67 
486.20 

£700/10/6 

$3,409.22 

*2.S0 

.67 
1.33 


$4-50 


The  ledger  would  contain  an  account  with  the  London  correspondent 
charged  and  credited  with  the  entries  in  the  Par  columns,  and  an  account 
with  Exchange,  showing  a  debit  of  $0-51  and  credits  of  $2.67  and  $4.50. 
The  sterling  column  is  a  memorandum  inventory  account  only.  The 
account  with  Exchange  would  indicate  a  profit  of  $6.66,  the  diflference  from 
the  profit  of  $6.74  shown  by  the  United  States  method  being  caused  by 
inventorying  the  balance  at  par  instead  of  at  cost.  Of  course,  the  postings 
would  be  made  in  daily  totals  in  a  bank  in  which  a  daily  trial  balance  is 
taken  off. 

The  balance  may  be  entered  at  cost  of  $4.86K,  or  $486.28  in  the 
sold  column  and  8  cents  in  the  credit  exchange  column,  which  will  make 
the  profit  $6.74,  the  same  as  by  the  other  method. 


Average  Date  of  Current  Account 

The  principles  already  explained  for  determining  the  average 
date  of  an  account  apply  when  the  account  is  kept  in  a  foreign 
coinage.     The  following  is  an  illustration. 

Illustration 

A  of  London,  in  current  account  with  B  of  New  York,  engages  an  account- 
ant to  prepare  a  statement,  to  be  mailed  to  B,  based  on  the  following  data: 

1914 
Debits: 

May  12 £750 

May  30 117 

June  12 340 

July     1 150 

Total  debits £1,357 


FOREIGN  EXCHANGE  129 

Credits: 

June  10 ^Soo 

June  30 300 

Total  credits 800 


Balance i    557 


Find  the  average  due  date  of  the  account  and  the  interest  at  5%  to 
July  I,  taking  365  days  to  the  year. 

Solution: 

Focal  date,  April  30 — 

Debits 

May  12 £  750  12  days  £9,000 

"    30 117  30     "  3,510 

June  12 340  43     "  14,620 

July     1 150  62     "  9,300 

£1,357  £36,430 

Credits 

June  10 £  500       41  days  £20,500 

"   30 300       61  "     18,300 

£  800  38,800 

Balances £  557  Debit  £2,370  Credit 

The  amount,  2,370  divided  by  557  equals  4-  But  since  the  balance 
of  the  account  is  a  debit  while  the  balance  of  the  products  is  a  credit,  the 
average  date  is  four  days  backward  from  April  30,  or  April  26. 

From  April  26  to  July  i  is  66  days.  Interest  on  £557  for  66  days  at 
5%  may  be  calculated  in  either  of  the  following  ways: 

I.  £557  X  .05  X  ^  =  £5-57  X    -  =  ^5-036 

£1  X   .036=      2osX.o36=     .72s 
isX     -72=     12  dX     .72  =  8.64  d 
Total  interest     £5/0/9 


130         MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 

2.  At  6%  •fS-S?    is  int.  for  6o  days  on  360  days  basis 

"  6%  .557 


£6.127  "    "      "    66     " 
"   1%  1.021 


n      u         u       a  II  u        ii  u 


"    5%  £5.106  "    " 

Reducing  to 
365  day  basis  .07    Subtract  1/73 


a       a         ic  II       (I 


£5.036  which  is  reduced  to  £5/0/9  as  above 

The  decimal  .036  can  also  be  reduced  as  follows: 

£1         =   240  d 

£  .036  =   240  X  .036,  or  9  d 

Conversion  of  Foreign  Branch  Accounts 

The  following  example  shows  how  foreign  branch  account 
balances  are  converted  to  domestic  values,  and  how  the  home 
office  and  foreign  branch  accounts  are  consolidated  in  the  period- 
ical statements. 

Illustration 

A  New  York  corporation  builds  a  plant  and  establishes  a  branch  in 
Liverpool,  England.  At  the  expiration  of  its  fiscal  period,  a  trial  balance 
is  forwarded  to  the  New  York  ofhce,  as  follows: 

Plant £250,000 

Accounts  receivable 187,500 

Expenses 25,000 

Inventory  (end  of  fiscal  period) 50,000 

Remittance  account 150,000 

Cash 12,500 

Accounts  payable £  87,500 

Income  from  sales 250,000 

New  York  office 33 7,500 

£675,000     £675,000 


FOREIGN  EXCHANGE  131 

A  trial  balance  of  the  New  York  books  on  the  same  date  is  as  follows: 

Capital  stock $2,500,000.00 

Patents $1,500,000.00 

Liverpool  account 1,640,250.00 

Remittance  account 729,281.25 

Expenses  at  New  York 25,000.00 

Cash 64,031.25 

$3,229,281.25     $3,229,281.25 


The  remittance  account  is  composed  of  four  60 -day  drafts  on  Liverpool 
for  £37,500  each,  which  were  sold  in  New  York  at  $4.85^^,  $4.86,  $4.86^^ 
and  $4.86^  respectively. 

Prepare  a  balance  sheet  of  the  New  York  books  after  closing  and  a 
statement  of  assets  and  liabilities  of  the  Liverpool  branch  consolidated  with 
the  New  York  books.  Close  the  books  at  the  rate  of  exchange  on  the  last 
day  of  the  fiscal  period,  which  is  $4.87 1<4,  conversion  of  remittances  to  be 
made  at  the  average  rate  for  the  four  bills. 

Solution:  Since  the  fixed  assets  are  not  subject  to  revaluation  at 
current  exchange  rates,  the  plant  cost  must  be  taken  out  of  the  New  York 
office  account,  which  is  subdivided  to  show  indebtedness  to  the  main  office 
for  fixed  assets  and  for  current  assets.  Since  the  cost  of  the  fixed  assets  is 
not  given,  we  must  assume  that  it  is  at  the  same  rate  as  the  rest  of  the 
New  York  account.  Dividing  the  Liverpool  account  balance  of  $1,640,250 
by  the  New  York  office  account  balance  of  £337,500  we  obtain  the  ex- 
change rate  of  4.86,  at  which  rate  the  plant  has  a  value  in  United  States 
money  of  $1,215,000.  The  reciprocal  accounts  are  then  subdivided  as 
follows : 

On  Liverpool  books: 

New  York  office — plant £250,000 

"        "        "     — current  account 87,500 

On  New  York  books: 

Liverpool   account — plant $1,215,000.00 

"         current  account 425,250.00 

The  next  step  is  to  close  the  Liverpool  books  into  the  New  York  office 
current  account  as  follows: 


132  MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 

New  York  Office  Current  Account 

Expenses £  25,000     Balance £  87,500 

Remittance  account  .  .  .         150,000     Income  from  sales 250,000 

Balance 162,500 


£337,500  £337,500 


Balance.  .  .  .• £162,500 

The  balance  sheet  of  the  Liverpool  branch  is  now  as  follows:  (Plant 
valued  at  cost,  $4.86;  current  assets  and  liabilities  valued  at  current  ex- 
change rate  of  $4.87}^). 

Assets 

Plant $4-86 

Accounts  receivable 4-^7/4 

Inventory 4-^7/^ 

Cash 4.87M' 


£250,000 

$1,215,000.00 

187,500 

913,503-75 

50,000 

243,625.00 

12,500 

60,906.25 

£500,000     $2,433,125.00 


Liabilities 

Accounts  payable $4-87K  £87,500  $    426,343.75 

New  York  office — plant  account  ....       4.86  250,000  1,215,000.00 

New  York  office — current  account —       A-^7%  162,500  791,781.25 


£500,000     $2,433,125.00 


On  the  New  York  books,  the  Liverpool  current  account  is  credited 
with  the  remittances,  and  a  charge  is  made  to  the  account  (offset  by  a 
credit  to  profit  and  loss)  sufficient  to  leave  the  account  with  a  balance  of 
$791,781.25,  the  balance  of  the  net  current  assets  in  Liverpool  at  $4.87>^. 
It  is  not  necessary  to  find  the  average  rate  of  the  remittance  account 
since  the  whole  account  is  used.  It  would  be  found,  by  dividing  $729,- 
281.25  by  150,000,  to  be  $4.86  3/16.  The  figures  to  be  used  are  then  found 
by  multiplying  £150,000  by  $4.86  3/16,  which  gives  $729,281.25,  an 
amount  which  we  already  have. 


FOREIGN  EXCHANGE 


133 


Liverpool  Current  Account 

Balance $    425,250.00     Remittances...     $    729,281.25 

Profit 1,095,812.50     Balance 791,781.25 


M, 521,062. 50 


51,521,062.50 


Balance $    791,781.25 

The  New  York  profit  and  loss  account  will  appear  as  follows: 

Profit  and  Loss 

New  York  expenses..  .$      25,000.00     Liverpool  Branch $1,095,812.50 

Net  profit  to  surplus. .  1,070,812.50 
$1,095,812.50 


$1,095,812.50 


This  profit  is  subject  to  a  further  charge  for  reduction  of  the  patents 
account. 

The  New  York  office  trial  balance  is  as  follows: 


Capital  stock 

Surplus 

Patents 

Liverpool  plant  account . . 
Liverpool  current  account 
Cash 


$2,500,000.00 
1,070,812.50 
$1,500,000.00 
1,215.000.00 
791,781.25 

64,031-25     

$3,570,812.50     $3,570,812.50 


The  balance  sheet  is  as  follows: 

Assets 
Fixed  assets: 

Patents 

Liverpool  Plant 

Current  assets: 

Accounts  receivable, 

Liverpool 

Inventory  

Cash — Liverpool $60,906.25 

Cash — New  York 64,031.25 


51,500,000.00 
1,215,000.00 


5    013.593-75 
243,625.00 


52.715,000.00 


124,937-50       1.282,156.25 
$3,997,156.25 


134         MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 

Liabilities 

Capital  stock $2,500,000.00 

Surplus 1,070,812.50    $3,570,812.50 

Accounts  payable,  Liverpool 426,343.75 

$3,997,156.25 

The  Liverpool  branch  profit  of  $1,095,912.50  is  made  up  as  follows: 

Income  from  sales £250,000 

Less  expenses 25,000 

Net  profit  on  sales £225,000,  at  $4.87/,^     $1,096,312.50 

Increase  of  current  exchange  rate  ($4. 8714^) 
over  original  valuation  ($4.86)  of  opening 
balance  of  current  account: 

£87,500  at  $4.87^ $426,343.75 

£8/, 500  at  $4.86 425,250.00  1,093.75 

$1,097,406.25 

Less  exchange  loss  on  remittances: 

£150,000  at  $4.87^  (current  rate) ....     $730,875.00 
£150,000  at   $4.861875    (average   con- 
version rate) 729,281.25  i, 593-75 

Net  profit $1,095,812.50 

Even  in  this  explanation  it  is  not  necessary  to  use  the  average  con- 
version rate. 


CHAPTER  XV 

LOGARITHMS 

Use 

Logarithms  are  a  special  compilation  of  figures,  used  to  reduce 
the  labor  of  multiplication,  division,  computation  of  powers  and 
extraction  of  roots.  After  explaining  the  method  of  employing 
logarithms,  their  theory  will  be  briefly  discussed. 

Every  number  has  its  corresponding  logarithm — but  a  knowl- 
edge of  the  method  of  determining  the  logarithms  of  various 
numbers  is  not  essential,  as  they  may  be  obtained  from  prepared 
tables.  The  logarithms  of  certain  numbers  are  stated  below  for 
use  in  illustrations.     For  a  more  complete  table,  see  Appendix. 

Table  of  Logarithms  of  Selected  Numbers 


Number 

Logarithm 

2 

•30103 

3 

.47712 

4 

.60206 

5 

.69897 

6 

•7781S 

7 

.84510 

8 

.90309 

9 

•95424 

lO 

1. 00000 

12 

1. 07918 

15 

I.I  7609 

16 

1. 20412 

30 

1.47712 

100 

2.00000 

160 

2.20412 

300 

2.47712 

1,000 

3.00000 

1,600 

3.20412 

3,000 

3-47712 

135 


136 


MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 


Multiplication 

Multiplication  is  accomplished  by  the  addition  of  logarithms, 
as  follows: 

1.  In  the  table,  find  the  logarithms  of  the  numbers  to  be 

multipHed. 

2.  Add  the  logarithms;  the  sum  is  the  logarithm  of  the 

desired  product. 

3.  In   the   table,   find   the   number  corresponding   to   the 

logarithm  obtained  in  2. 

The  following  examples,  though  more  easily  performed  with- 
out logarithms,  will  illustrate  the  steps  of  the  process; 

Example 

2X3=? 


Solution: 

1.  Table  shows 

2.  Sum  of  logarithms 


Number       Logarithm 

2  .30103 

3  -47712 


.77815       log  of  product 


3.  Table  shows  .77815  is  log  of  6,  the  product 

Example 

2X3X2=? 


Solution: 

1.  Table  shows 

<(  a 

2.  Sum  of  logarithms 


Number  Logarithm 

2  -30103 

3  -47712 
2  -30103 


1.07918       log  of  product 


3.  Table  shows  1.07918  is  log  of  12,  the  product 


LOGARITHMS  137 

Without  going  beyond  the  selected  table,  the  following  multipli- 
cations may  be  performed  by  using  logarithms: 

5X6=?  4X3=?  3X2X5=? 

Division 

Division  is  accomplished  by  the  subtraction  of  logarithms 
as  follows: 

1.  In  the  table  find  the  logarithms  of  the  dividend  and  the 

divisor. 

2.  Subtract  the  logarithm  of  the  divisor  from  the  logarithm 

of  the  dividend;  the  difference  is  the  logarithm  of  the 
desired  quotient. 

3.  In  the  table,  find  the  number  corresponding  to  the  loga- 

rithm obtained  in  2. 

Example 
6-^2=? 

Solution: 

Number     Logarithm 

1.  Table  shows: 

Dividend 6  -77815 

Divisor 2  -30103 

2.  Difference  of  logarithms -47712       log  of  quotient 

3.  Table  shows  .47712  is  log  of  3,  the  quotient 

The  following  divisions  may  be  performed  by  logarithms  with 
the  aid  of  the  selected  table: 

8-i-2  10  4-5  16-J-4  300  -T-  30 

Calculating  Powers 

A  power  of  a  number  i3  the  product  obtained  by  using  that 
number  repeatedly  as  a  factor.     The  square  of  a.  number  is  the 


138         MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 

second  power;  the  cube,  the  third  power,  etc.  The  exponent 
of  the  power  indicates  the  number  of  times  the  factor  is  to  be 
used. 

For  instance  3^  means  the  second  power  of  3;  or  3  squared; 
the  number  used  as  a  factor  is  3.  The  superscript  number  ""  is 
the  exponent  of  the  power,  showing  that  3  is  to  be  used  twice  as 
a  factor. 

Thus,  3'  =  3  X  3  =  9 

A  number  is  raised  to  a  power  by  logarithms  as  follows: 

1 .  In  the  table  find  the  logarithm  of  the  number  to  be  raised 

to  a  power. 

2.  Multiply  the  logarithm  by  the  exponent  of  the  power;  the 

product  is  the  logarithm  of  the  power. 

3.  In  the  table  find  the  number  corresponding  to  the  loga- 

rithm obtained  in  2. 

Example  i 

3'=  ? 

Solution: 

Number      Logarithm 

1.  Table  shows 3  -47712 

2.  Multiply  by  exponent  of  power 2 

.95424      log  of  power 

3.  Table  shows  .95424  is  log  of  9,  the  second  power  of  3. 

Example  2 

2"     =    ? 

Solution: 

Number      Logarithm 

1 .  Table  shows 2  -30103 

2.  Multiply  by  exponent  of  power 4 

1. 204 1 2  log  of  power 

3.  Table  shows  1.204x2  is  log  of  16,  the  fourth  power  of  2. 


LOGARITHMS  ^39 

Roots 

A  root  is  a  number  which,  used  repeatedly  as  a  factor,  will 

produce  a  given  power. 

The  square  root  of  9  is  3,  because  3  used  twice  as  a  factor 

produces  9.  ,       •  r  ^f^^ 

The  fourth  root  of  16  is  2,  because  2  used  4  times  as  a  factor 

^'""ThTroot  of  a  number  is  expressed  by  writing  the  number 
under  the  radical  sign;  the  index  figure  indicates  the  root  to  be 

extracted. 

The  square  root  of  9  is  expressed  thus: 

The  superscript  ^  is  the  index  of  the  root. 

Any  root  of  a  number  may  be  extracted  by  logarithms  as 

follows: 

I.  In  the  table  find  the  logarithm  of  the  number,  the  root  of 
which  is  to  be  extracted. 

2  Divide  the  logarithm  by  the  index  of  the  root;  the  quo- 
tient is  the  logarithm  of  the  desired  root. 

3.  In  the  table  find  the  number  corresponding  to  the  loga- 
rithm obtained  in  2 . 

Example  i 


Solution: 


I.  Table  shows  log  of  9  to  be  .95424 

2    Divide  by  index  of  root:     .954^4  -  ^  =  .47712,  log  of  root 

;.  Table  shows  .47712  is  log  of  3,  the  square  root  of  9 


Example  2 

<r^=  ? 


140         MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 

Solution: 

1.  Table  shows  log  of  i6  to  be  i. 2041 2 

2.  Divide  by  index  of  root:     i. 20412  -^  4  =  .30103,  log  of  root 

3.  Table  shows  .30103  is  log  of  2,  the  fourth  root  of  16 

Nature  of  Logarithms 

Logarithms  are  exponents  of  the  base  10.  The  logarithm 
of  any  number  shows  how  many  times  10  must  be  used  as  a  factor 
to  produce  that  number.  This  is  easily  seen  in  the  case  of  num- 
bers which  are  exact  powers  of  10.  The  selected  table  shows  the 
following : 

Number  Logarithm 
10  1. 00000 

100  2.00000        (10  X  10  =  100) 

1000  3.00000        (10  X  10  X  10  =  1000) 

Numbers  which  are  the  exact  powers  of  10  have  logarithms 
which  are  whole  numbers. 

Most  numbers  which  are  not  exact  powers  of  10  have  loga- 
rithms which  are  mixed  numbers,  composed  of  two  parts:  an 
integer,  and  a  decimal  fraction. 

The  following  are  taken  from  the  selected  table: 


Dumber 

Logarithm 

10 

1. 00000 

12 

1.07918 

15 

I.I  7609 

16 

1. 20412 

30 

1.47712 

100 

2.00000 

160 

2.20412 

1,000 

3.00000 

1,600 

3.20412 

The  Characteristic  and  the  Mantissa 

The  two  parts  of  a  logarithm  are:  the  characteristic,  which 
is  an  integer,  and  the  mantissa,  which  is  the  decimal  part. 


LOGARITHMS  I4I 

The  mantissa  is  determined  by  the  figures  of  the  number; 
(initial  and  final  zeroes  in  the  number  do  not  affect  the  mantissa). 
Thus  the  logarithms  of  the  numbers  16,  160,  1600,  etc.,  all  have 
the  same  mantissa,  .20412. 

The  characteristic  indicates  the  exponent  of  the  largest  power 
of  10  contained  in  the  number. 

To  illustrate: 


Largest  Power 

OF 

10 

Exponent 

VT  UMBER 

CONT. 

\INED   IN   N 

UMBER 

OF 

Power 

Logarithm 

16. 

10 

I 

I. 20412 

160. 

100 

2 

2.20412 

1600. 

1000 

3 

3.20412 

16000. 

1 0000 

4 

4.20412 

It  will  be  noted  that  the  characteristic  of  the  logarithm  is  always 
one  less  than  the  number  of  figures  at  the  left  of  the  decimal 
point  in  the  number. 

Figures  .a.t  Left 
Number     of  Decimal  Point       Characteristic      Logarithm 

16  2  I  1. 20412 

160  3  2  2.20412 

1600  4  3  3.20412 

16000  5  4  4.20412 

Tables  of  Logarithms 

Referring  to  the  table  of  logarithms  in  the  Appendix  it  will 
be  noted  that  only  the  mantissas  are  given;  in  forming  logarithms 
the  mantissa  is  taken  from  the  table,  and  the  characteristic  is 
determined  by  the  location  of  the  decimal  point  in  the  number. 
The  following  lines  are  taken  from  the  table  in  the  Appendix: 

No.        01  23456789 

160  20412   20439   20466   20493   20520   20548   20575   20602   20629   20656 

161  20683   20710  20737   20763   20790  20817   20844   20871    20898   20925 

The  heavy  figures  at  the  left  and  at  the  top  indicate  the  se- 
quence of  figures  in  the  number. 


142         MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 

The  smaller  numbers  in  the  body  of  the  table  indicate  the 
mantissas.     For  instance : 

Line  i6o,  Column  o,  shows  20412,  which  is  the  mantissa  of 
16,  160,  1,600,  16,000,  etc. 

Line  160,  Column  4,  shows  20520,  which  is  the  mantissa  of 
16.04,  160.4,  i)6o4,  16,040,  etc. 

Line  161,  Column  9,  shows  20925,  which  is  the  mantissa  of 
16.19,  161. 9,  1,619,  16,190,  etc. 

Applying  the  method  explained  in  the  preceding  section  and 
using  the  sequence  of  figures  1,619  as  an  illustration,  the  char- 
acteristic is  ascertained  and  prefixed  as  follows: 

Figures  at  Left 
Number      of  Decimal  Point      Characteristic      Logarithm 

16,190  5  4  4-20925 

1,619  4  3  3-20925 

161. 9  3  2  2.20925 

16.19  2  I  1.20925 

Characteristics  of  Logarithms  of  Numbers  between  i  and  10 

From  the  preceding  illustration  it  will  be  noted  that  as  the 
decimal  point  is  moved  one  place  to  the  left  the  characteristic 
decreases  by  i . 

The  table  is  continued  in  the  following: 

Figures  at  Left 
Number       of  Decimal  Point      Characteristic      Logarithm 

16.19  2  I  1.20925 

1. 619  I  o  0.20925 

or  20925 

Thus  it  is  seen  that  the  characteristic  is  o  when  the  number  lies 
between  i  and  10,  which  is  in  accordance  with  the  general  rule 
that  the  characteristic  is  one  less  than  the  number  of  figures  at  the 
left  of  the  decimal  point. 


LOGARITHMS  I43 

Examples 

Determine  the  characteristics  and  state  the  logarithms  of  the  follow- 
ing numbers: 

25,000  2.5  492 

2,500        37,010  7,635 

250  370.1  200 

25  3.701         4,005 

Use  of  the  Characteristic  in  Pointing  Off  Results 

If  the  characteristic  of  a  logarithm  is  one  less  than  the  number 
of  figures  at  the  left  of  the  decimal  point  in  the  number,  then  the 
figures  at  the  left  of  the  decimal  point  in  the  number  must  be  one 
more  than  the  characteristic. 

Referring  to  the  brief  table  on  page  141,  20790  is  the  mantissa 
of  the  number  1,614;  the  characteristic  preceding  this  mantissa 
would  determine  the  location  of  the  decimal  point.    To  illustrate : 


Places 

at 

Left 

3GARITHM 

Character: 

STIC 

OF  Decimal 

Point 

Number 

4.20790 

4 

5 

16,140. 

3.20790 

3 

4 

1,614. 

2.20790 

2 

3 

161. 4 

1.20790 

I 

2 

16.14 

.20790 

0 

I 

1. 614 

The  following  examples  are  given  to  illustrate  the  relation 
between  the  characteristic  and  the  position  of  the  decimal  point. 

Illustration 

24  X  3-6  =  ? 
Solution  (Refer  to  table  for  mantissas) : 
I.  Determine  logarithms: 

log  24:     mantissa  .38021 

characteristic      i 
logarithm  i. 38021 

log  3.6:     mantissa  55630 

characteristic     o 
logarithm  0.55630 


144         MATHEMATICS  OP  ACCOUNTING  AND  FINANCE 

2.  Add  logarithms: 

1. 3802 1     log  24 
0-55630       "  3-6 


1.9365 1     log  of  product 

3.  Determine  number  corresponding  to  the  logarithm  1.93651. 
Mantissa  .93651  is  found  in  the  table  thus: 


No. 

0 

864 

93651 

and  represents  sequence  8640. 

Characteristic  i  indicates  that  there  are  to  be  two  places  at  the  left 
of  the  decimal  point  in  the  product,  86.4. 


Illustration 
9.46  ^  4.3  = 


Solution: 


I.  log  9.46:  mantissa  -97589 

characteristic     o 
logarithm  0.97589 


log  43: 


mantissa  -63347 

characteristic     o 
logarithm  0.63347 

0.97589  log  of  dividend 

0-63347    "     "  divisor 

0.34242    "     "   quotient 

.34242  is  mantissa  of  figures  2200 
Characteristic  o;  one  figure  at  left  of 
decimal  point;  quotient  2.2 


LOGARITHMS 


145 


Examples 

ind  the  value  of  the  following: 

I.   23     X  IS 

7- 

8^ 

2-  35     X  2.5 

8. 

4^ 

3.   II. 4  X  7 

9- 

1.2^ 

4.  392   -^  8 

10. 

V  729 

5.  952   -^  28 

II. 

^64 

6.  9.38  -J-  1.4 

12. 

V^  5-29 

Logarithms  of  Numbers  Smaller  than  i 

The  following  numbers  with  their  logarithms  have  been  used 
to  show  that  moving  the  decimal  point  in  a  number  one  place  to 
the  left  reduces  by  i  the  characteristic  of  the  logarithm. 


Number 
16,190. 
1,619. 
161. 9 
16.19 
1. 619 


Figures  at  Left 
OF  Decimal  Point 

S 
4 
3 

2 


Characteristic 
4 
3 

2 

I 


Logarithm 
4.20925 
3.20925 
2.20925 
1.20925 
0.20925 


If  the  decimal  point  is  again  moved  one  place  to  the  left,  we 
have: 


.1619 


1.20925 


In  other  words,  if  the  number  is  less  than  i ,  the  characteristic 
of  the  logarithm  is  a  negative  quantity.  The  minus  sign  is  placed 
above  the  negative  characteristic  to  show  that  it  applies  to  the 
characteristic  only,  the  mantissa  being  a  positive  quantity.  The 
logarithm  1.20925  may  also  be  written  9.20925-10. 

The  negative  characteristic  is  one  greater  than  the  number  of 
zeroes  between  the  decimal  point  and  the  first  significant  figure 
in  the  number. 

In  the  following,  the  table  of  numbers  smaller  than  i  and  their 
logarithms  is  continued : 


146         MATHEMATICS  OP  ACCOUNTING  AND  FINANCE 


^J  UMBER 

Zeros  Between 

Decimal  Point  and 

First  Significant 

Figure 

Negative 
Characteristic 

Logarithm 

.1619 

0 

—  I 

1.20925  or  9.20925  —  10 

.O161Q 

I 

—  2 

2.20925  or  8.20925  —  10 

.00 1 6 1 9 

2 

-3 

3.20925  or  7.20925  —  10 

etc. 

Examples 

Determine  the  characteristics  by  inspection;  refer  to  the  table  for  the 
mantissas;  and  state  the  logarithms  of  the  following  numbers: 

.346  .0346       .00346        .000346 

.0201       .0031       .29  .0101  .000003 

Use  of  Negative  Characteristic  in  Pointing  Off  Answers 

When  the  logarithm  obtained  by  a  computation  has  a  nega- 
tive characteristic,  it  indicates  that  the  corresponding  number  is 
smaller  than  i ;  that  is,  it  is  a  decimal  fraction.  Point  off  the 
number  by  placing  between  the  decimal  point  and  the  first  sig- 
nificant figure  one  fewer  zeros  than  the  number  of  the  character- 
istic thus : 


Negative 

ogarithm 

Characteristic 

Zeros 

Number 

3.20925 

-3 

2 

.001619 

2.20925 

—  2 

I 

.01619 

1.20925 

—  I 

0 

.1619 

Examples 

Determine  the  numbers  corresponding  to  the  following  logarithms: 
2.98005         3.88969         1.99782         0.71642 

Computing  with  Logarithms  Having  Negative  Characteristics 

The  following  are  illustrations  of  the  use  of  logarithms  with 
negative  characteristics,  in  multiplication,  division,  raising  to 
powers,  and  extraction  of  roots. 


Multiplication 

Illustration 

021  X  .000451  = 

? 

Solution: 

Number 

Logarithm 

.021 

2.32222  or 

8.32222  — 

-10 

.00045 1 

4.65418  or 

6.65418- 

- 10 

Sum  of  logs 

6.97640  or 

14.Q7640- 

-  20 

147 


.97640  is  the  mantissa  of  the  figures  9471 

Characteristic  6;  five  zeroes  in  answer;  product,  .000009471 

Division 

Illustration  i 

.0341  -f-  .31  =  ? 

Solution  : 

Number       Logarithm 

Dividend 0341  2.53275 

Divisor 31  i-49i36 

Logarithm  of  quotient i. 04139 


.04139  is  the  mantissa  of  the  figures  11 
Characteristic  7;  no  zeroes;  quotient  .11 

Illustration  2 

.341  -j-  .00031  =  ? 

Solution: 

Number        Logarithm 

Dividend 34i  1-53275 

Divisor 00031  4-4QI36 

Logarithm  of  quotient 3.04139 

.04139  is  mantissa  of  11 

Characteristic  3  is  positive;   point   off   four 
places  from  the  left;  quotient  1,100 


148         MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 

Illustration  3 

34.1  -f-  .31  =  ? 

Solution: 

Number  Logarithm 

Dividend 34.1  1-53275 

Divisor 031  2.49136 

Logarithm  of  quotient 3.04130 

(Note:  subtracting  —  2  is  the  same  as  adding 

+  2) 
.04139  is  the  mantissa  of  11 
Characteristic  3 ;  put  decimal  point  four  places 
from  left;  quotient  1,100 

Powers 

Illustration 

.034^  =  ? 

Solution: 

Number  Logarithm 

.034  2.53i4« 

Multiply  by  exponent 2 

Logarithm  of  power 3.06296 

Note:  the  3  characteristic  is  the  result  of  the  following: 

Characteristic  Mantissa 

Logarithm 2  -53148 

Multiply  by 2  2 

4  1.06296 

The  difference  between  4  and  i  is  3,  as  above 

.06296  is  mantissa  of  1156 

Characteristic  3;  two  zeroes  required;  hence  .034^  =  .001156 


LOGARITHMS 


149 


Roots 


Illustration 

-V/.000729 


=  7 


Solution: 


Number 

.000729 


Logarithm 

4.86273 


This  logarithm  is  to  be  divided  by  3;  but  if  —4  is  divided  by  3,  the 
quotient  is  —  i,  and  —  2>2>?)  "I"  to  carry.  This  negative  remainder  causes 
confusion  in  the  division  of  the  positive  mantissa.  Hence  the  procedure 
below  is  followed: 


Add  and  subtract  20* 
Divide  by  3 


Number    Logarithm 
000729      4.86273  or    6.86273  —  10 
20.  —  20 


3  )  26.86273  -  30 
8.95424  —  10 
)r        2.95424 


.95424  is  the  mantissa  of  9 

Characteristic  2;    one    zero;    hence    -y^  .000729    =  .09 

*2o  is  used  to  obtain  the  —30,  which  is  divisible  by  3 


Examples 


Compute  the  following: 

I.  .031  X  .3 

2 

.0045  X  .00021 

3 

.0036  X  2.2 

4 

.000011  X  56000 

5 

.0023  X  42000 

6 

.006  X  6.5 

7 

.038  X  .0003 

8 

504  -f-  24 

9 

.368  -r    23 

10 

.00351  -^  .027 

II. 

390  H- 

•15 

12. 
13- 

.0994  - 
28^ 

-  71 

14. 

1.3^ 

15- 
16. 

17- 

.9^ 
.003  <^ 

18.  V.0625 

19.  V. 000343 

20.  4^  X  \/.ooi6 


150         MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 

Determining  Mantissas  by  Interpolation 

Tables  of  logarithms  are  limited  in  scope  and  do  not  contain 
the  mantissas  of  all  possible  numbers.  For  numbers  beyond  the 
scope  of  the  table,  the  mantissa  may  be  approximated  by 
interpolation. 

Illustration 

What  is  the  logarithm  of  18565? 

Solution:  Number      Mantissa 

The  table  shows 18570  .26S81 

" 18560  .26858 

Difference 10  .00023 

If  a  difference  of  10  in  the  numbers  causes  a  change  of  .00023  in  the 
mantissa,  a  difference  of  5  will  cause  a  change  of  approximately  V.o  of 
.00023  in  the  mantissa. 

Now,  .5  of  .00023    =  .000115 

Then  .26858,  mantissa  of  18560 

plus         .000115 

sum         .268695,  mantissa  of  18565  (approximate) 

Logarithm,  4.268695 

Examples 

By  interpolation  determine  the  logarithms  of  the  following  numbers: 

3642      1209      47629        758263 
25.69     3.479     .004682     32.0046 

Determining  Numbers  by  Interpolation 

When  a  computation  results  in  a  mantissa  not  to  be  found  in 
the  table,  the  number  corresponding  to  the  mantissa  may  be 
determined  approximately  by  interpolation. 


Solution: 


LOGARITHMS  151 

Illustration 
15^=  ? 


Number      Logarithm 
IS  I. 17600 


Multiply  by  exponent  of  power 4 


Logarithm  of  1 5  "* 4. 70436 


The  mantissas  larger  and  smaller  than  .70436  shown  in  the  table  are: 

Number     Mantissa 


5063 
5062 

I 

.70441 
•70432 

Differences       .... 

.00009 

The  mantissa  oils'*     is 

"    5,062  is  ...  . 

.70436 
•70432 

Difference 

.00004 

If  an  increase  of  .00009  in  the  mantissa  represents  an  increase  of  i  in 
the  number,  an  increase  of  .00004  in  the  mantissa  represents  an  increase 
of  approximately  4/9  of  i,  or  .444  +  in  the  number. 

Hence  .70432  Mantissa  of  5062 

add  interpolation  for     .00004  444  + 


Result  .70436  mantissa  of  5062444  + 

4.70436  is  the  logarithm  of  50624.44  or  15'' 
Exact  answer  is  50,625. 

Examples 

What   are   the   approximate  numbers  corresponding  to  the  following 
logarithms? 

2.76354     4^94965 
3.42345     2.4166572 
1.82629     3.57927 


152 


MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 


Illustration 

What  is  the  value  of  1856^? 


Solution: 


The  logarithm  of  1,856,  is 3.26858 

Multiply  by  exponent  of  power 2 


Logarithm  of  1856^ 6.53716 


Table  shows:  Number 

Larger  mantissa 3445 

Smaller       "        3444 

Differences i 


M.-VNTISSA 

53719 
53706 


Mantissa  of. 


1856^ 
3444 


Difference. 


00013 


53716 
53706 


10/13  of  I  =  .769  approximately 

Hence 3444 

Plus 760 


•53706 
.00010 


Sum 3444769     .53716 


Then  3444769  is  the  number  represented  by  6.537148 

1856^  is  accurately 3444736 

Approximated  by  logarithms 3444769 

Error 3^ 

If  the  problem  had  been  18.56'' 

The  accurate  power  would  be 344.4736 

Approximate  power  (by  logs) 344.4769 


Error . 


■0033 


This  example  shows  that  fairly  accurate  results  may  be 
obtained  by  interpolation,  but  absolutely  accurate  work  requires 
tables  extended  to  show  the  mantissas  of  all  numbers  required 
by  the  problem. 


LOGARITHMS  153 

Examples 


What  are  the  values  of: 


1.  24  X  18 

2.  360  X  800 

3.  32.6  X  3 

4.  .27  X  .71 

5.  864  X  .00321 

6.  2462  X  3278 

7.  .00964  X  3425 

8.  34400 -^  43  21.  </  .001833316 


9.  9280  -J-   232  22.    -y^  .0000000032 

10.  1 1000  -T-  12  23.  '/  768 

11.  5893  -^  830  M   12 

12.  754.314-  6234  24.  -'      81^ 

13.  17^  ^     ^ 


CHAPTER  XVI 
SIMPLE   AND    COMPOUND   INTEREST 

Simple  Interest — Methods  of  Calculating 

Interest  is  the  income  or  expense  arising  from  the  use  of  money 
or  credit  and  the  increase  in  investment  or  indebtedness  resulting 
therefrom. 

When  only  the  original  indebtedness  bears  interest,  the  prin- 
ciples of  simple  interest  apply;  when  the  interest  increase  of  the 
indebtedness  also  bears  interest,  the  principles  of  compound 
interest  apply. 

There  are  a  number  of  methods  of  computing  simple  interest. 
Those  given  below  are  among  the  easiest.  These  methods  are 
based  upon  a  rate  of  6%,  since  that  is  a  common  rate  and  is 
besides  a  factor  of  1 2,  the  number  of  months  in  a  year,  and  of  30, 
the  number  of  days  in  a  month.  Adjustments  may  be  easily 
made  for  other  rates.  The  methods  described  are  also  based 
on  360  days  to  a  year.  Corrections  may  be  made  to  adjust  the 
result  to  a  basis  of  365  days. 

A  rate  of  6%  (.06)  per  year  is  equivalent  to  a  rate  of  1% 
(.01)  for  2  months  or  60  days  (one  sixth  of  a  year).  As  .01  of  a 
number  is  computed  by  moving  the  decimal  point  two  places  to 
the  left,  the  interest  on  any  principal  at  6%  for  2  months  or  60 
days  may  be  computed  by  moving  the  decimal  point  two  places 
to  the  left.  Since  6  days  are  one  tenth  of  60  days,  interest  for  6 
days  may  be  computed  by  moving  the  decimal  point  three  places 
to  the  left.  And  since  600  days  are  ten  times  60  days,  interest 
at  6%  for  600  days  may  be  computed  by  moving  the  decimal 
point  one  place  to  the  left. 

154 


SIMPLE  AND  COMPOUi\D  INTEREST  155 

Thus,  to  find  the  interest  at  6%  on  any  principal  for 

6  days,  point  off  in  the  principal  3  decimal  places  to  the  left 
60    "  "     "      "    "         "  2       "  "       "     "      " 

600    "  "     "      "    "         "  I       "  place    "     "      " 

6000    "  the  interest  is  the  same  as  the  principal 

For  instance,  the  interest  at  6%  on  $8,245.75  for 

6  days  is         $8.24575  or      $8.25 
60    "       "       $82.4575    or    $82.46 
600    "       "     $824,575      or  $824.58 
6,000    "       "  $8,245.75 

The  time  is  frequently  a  number  of  days  which  maybe 
separated  into  parts,  each  of  which  is  a  fraction  or  multiple  of  6, 
60,  600,  or  6,000.  In  such  cases  the  interest  may  be  computed  by 
finding  the  total  of  the  interest  for  the  component  time  parts. 

Illustration 

Find  the  interest  on  $726.32  for  88  days. 

Solution: 

$  7.2632     =  interest  for  60  days 

2.4211     =         "        "    20  "      (Mof  60) 
.72632  =         "        "       6  " 
.24211  =         "        "       2  "      {Hoi  6) 


$10.65273 


If  the  days  are  not  convenient  fractions  or  multiplies  of  6,  60, 
etc.,  the  principal  may  be  such  a  fraction  or  multiple,  in  which 
case  principal  and  time  may  be  interchanged  thus: 

Illustration 
Find  the  interest  on  $3,600  for  37  days. 

Solution:  This  is  equivalent  to  interest  on  $37  for  3600  days.  Then 
$3.70  (pointing  off  one  place;  the  interest  for  600  days,  multiplied  by  6 
equals  $22.20  the  interest  on  $37  for  3,600  days,  or  the  interest  on  $3 ,600 
for  37  days. 


156         MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 

When  neither  of  these  methods  can  be  utiHzed,  the  procedure 
is  as  follows : 

Point  off  three  places  in  the  principal — Result:  Interest  for 

six  days. 
Multiply  by  number  of  days — Result:  Interest  for  six  times 

the  number  of  days. 
Divide  by  6 — Result:  Interest  for  given  number  of  days. 

Illustration 

Find  the  interest  on  $827.52  for  119  days  at  6%. 

Solution: 

$     .82752  interest  for    6  days 
119 


$98.47488      "  "  6  X  119  days 

$98.47488 -^  6  =  $16.41248      "  "  119  days 

Rates  Other  than  6% 

To  calculate  simple  interest  when  the  rate  is  other  than  6%, 
the  interest  at  6%  is  found  and  the  result  is  adjusted  to  the  given 
rate. 

Illustration 

What  is  the  interest  on  $827.52  for  119  days  at  4>^%? 

Solution:  The  interest  at  6%  has  already  been  computed  being 
$16.41248.  Since  4>2%  is  yi  of  6%,  the  interest  at  4^%  is  found  as 
follows : 

$16.41248  interest  at  6     % 

4.. 03 1 2        "        "  iK%(><of6%) 

$12.30936     "     "  Ay2% 


SIMPLE  AND  COMPOUND  INTEREST  157 

365  Day  Basis 

To  adjust  to  a  basis  of  exact  interest,  365  days  to  a  year,  the 
interest  is  computed  on  a  360  day  basis  as  explained,  and  the 
result  is  decreased  y^j  of  itself.  The  5  days  difference  is  773  of 
365  days,  and  since  interest  is  a  computation  which  may  be 

expressed 

Principal  X  Rate  X  Days 

Number  of  days  in  Year 
the  use  of  360  (a  denominator  which  is  ^j ^^  too  small)  will  result 
in  the  interest's  being  y.3  too  large. 

Illustration 

What  is  the  exact  interest  on  $827.52  at  4>^%  for  119  days? 

Solution:  The  interest  at  ^Yi^o  on  a  360  day  basis  has  already  been 

computed,  being $12.30936 

Deduct  7^3  of  $12.30936 .16862 

Exact  interest  on  365  day  basis $12.14074 

Partial  Payments 

There  are  two  methods  of  calculating  interest  on  a  debt  on 
which  partial  payments  are  made.  The  one  in  common  use  among 
business  men  and  known  as  the  Merchants'  Rule,  consists  in 
calculating  the  interest  on  the  principal  sum  from  its  date  to  the 
date  of  settlement  and  a  similiar  calculation  on  the  partial  pay- 
ments, the  difference  between  the  total  interest  on  the  respective 
sides  being  the  net  interest  due.  On  small  sums  and  for  short 
periods  this  is  accurate  enough  for  all  practical  purposes. 

The  other,  called  the  United  States  Rule,  gives  precedence 
to  the  interest  due  at  the  time  of  each  payment,  and  requires  that 
each  payment  shall  be  first  applied  to  the  liquidation  of  the  inter- 
est then  due,  only  the  remainder  after  the  interest  is  deducted 
being  applicable  to  the  reduction  of  the  principal.  If  the  pay- 
ment is  not  equal  to  the  interest  then  due,  it  is  applied  in  reducing 
the  interest,  but  the  excess  interest  is  not  added  to  the  principal, 


158         MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 

as  this  would  be  compounding  it,  but  it  is  carried  down  and  added 
to  the  interest  to  be  deducted  from  the  next  payment.  This 
method  is  made  legal  by  statute  in  nearly  every  state. 

An  example  of  the  two  methods  will  show  the  difference 
between  them,  30  days  to  the  month  being  used  for  convenience. 

Illustration 

Jan.  I.     Original  amount         $600     March  i,  payment         $200 

May      I,         "  200 

June      I,         "  100 

Interest  is  to  be  charged  at  6%.     Required — amount  due  July  i 
Solution: 

By  the  Merchants'  Rule : 

Interest  on  $600  for  6  months  is $18.00 

"  "     200     "4        "        " $4.00 

"  "     200     "2        "        " 2.00 

"  "     100     "  I        "        " 50  6.50 

Interest  due  July  i $11.50 

Unpaid  principal 100.00 

Total  due  July  i $111.50 

By  the  United  Stales  Rule: 

Original  debt $600.00 

Payment  March  i $200.00 

Less  interest  on  $600  for  2  months 6.00       194.00 

$406.00 

Payment  May  i $200.00 

Less  interest  on  $406  for  2  months 4.06       195.94 

$210.06 

Payment  June  i $100.00 

Less  interest  on  $210.06  for  i  month.  .  .  1.05         98.95 

$111. II 

Interest  on  $111. 11  for  i  month .56 

Total  due  July  i $111.67 


SIMPLE  AND  COMPOUND  INTEREST  159 

While  the  difference  of  1 7  cents  in  the  results  obtained  by  the 
two  methods  is  negligible,  it  is  otherwise  when  the  amounts  are 
large  and  the  time  is  long.  The  following  example  is  taken  from 
the  accounts  of  an  estate  that  was  not  settled  for  thirty  years. 
It  was  virtually  as  follows : 

Illustration 

An  amount  of  $36,000  was  due  one  of  the  heirs  of  an  estate,  who  was  to 
receive  6%  interest  until  paid. 

January  i,  1886,  legacy $36,000 

"         I,  1 89 1,  payment 12,000 

"         I,  1896,          "      12,000 

"         I,  1906,          "      12,000 

"         I,  1911,          "      19,000 

The  administrator,  using  the  commercial  rule,  agreed  to  settle  the 
account  as  of  January  i,  1916,  as  follows: 

Original  amount $  36,000 

Interest  for  30  years  at  6% 64,800 

$100,800 

Less  total  cash  paid $55,000 

Interest  on  $12,000  for  25  years 18,000 

Interest  on  $12,000  for  20  years 14,400 

Interest  on  $12,000  for  10  years 7,200 

Interest  of  $19,000  for  5  years 5, 700       100,300 

Amount  due  January  i,  1916 $500 

The  heir  refused  the  settlement  and  made  his  claim  under  the  law  of  the 
state  (Illinois)  as  follows: 

Original  amount $36,000.00 

January  i,  1891,  payment $12,000.00 

Interest  5  years  on  $36,000.00. .  .  10,800.00         1,200.00 

$34,800.00 

January  i,  1896,  payment $12,000.00 

Interest  5  years  on  $34,800.00. .  .  10,440.00         1,560.00 

$33,240.00 


I60         MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 

January  i,  1906,  payment $12,000.00 

Interest  10  years  on  $33,240.00..  19,944.00 


Interest  carried  forward $  7,944.00 


January  i,  191 1,  payment $19,000.00 

Interest  5  years  on  $33,240.00. . .    $9,972.00 

Interest  brought  forward 7,944.00      17,916.00        1,084.00 


$32,156.00 


January     i,     1916,     interest     5     years 

on  $32,156.00 9,646.80 


Amount  due  January  i,  1916..  $41,802.80 


It  can  readily  be  seen  that  a  little  knowledge  of  correct  principles  was 
a  valuable  asset  to  the  heir. 


Compound  Interest 

The  principles  of  interest  are  involved  in  the  computations 
which  an  accountant  may  be  required  to  make  in  connection 
with  such  matters  as  bond  premium  and  discoimt,  leasehold 
premiums,  depreciation  and  sinking  funds.  These  calculations 
involve  not  only  compound  interest  and  the  amount  of  a  given 
principal  at  compound  interest,  but  also  the  more  complex 
problems  of  annuities. 

All  scientific  computations  of  interest  on  an  indebtedness  or  an 
investment  extending  over  more  than  one  period  of  time  must  be 
based  on  compound  interest.  This  is  because  the  indebtedness 
or  investment  increases  with  the  lapse  of  time.  If  only  simple 
interest  is  charged,  interest  is  earned  only  on  the  original  invest- 
ment instead  of  on  the  investment  for  each  period.  For  instance, 
if  $100  is  invested  at  6%,  the  investment  at  the  beginning  of  the 
first  year  is  $100.  It  increases  $6  during  the  first  year  and 
amounts  at  the  end  of  the  first  year  to  $106. 

Since  the  investment  has  now  increased  to  $106,  interest 


SIMPLE  AND  COMPOUND  INTEREST  l6l 

should  be  earned  thereon.  If  simple  interest  only  is  charged, 
$6  will  be  earned  during  the  second  year  on  an  investment  of 
$io6.     This  is  at  the  rate  of  only  5.66  +%. 

Scientific  computations  of  interest  are  based  on  the  sup- 
position that  the  increase  arising  from  interest  is  re-invested. 
This  is  equitable  in  theory,  for  if  the  interest  of  one  period  in- 
creases the  investment  at  the  close  of  that  period,  the  investor 
should,  during  the  next  period,  earn  interest  on  the  increased 
investment.  To  maintain  the  agreed  interest  rate  during  a 
series  of  periods,  computations  must  be  made  on  the  basis  of 
periodical  compounding. 

Symbols 

The  following  standard  symbols  by  Sprague  and  Perrine, 
will  be  used  throughout  this  book: 

I    =  $1,  £1,  or  any  other  unit  of  value 

i    =  the  rate  of  interest  for  a  single  period 

«  =  an  indefinite  number  of  periods 

a   =  the  amount  of  $1  for  a  given  time  at  a  given  rate 

/   =  the  compound  interest  on  $1  for  a  given  time  at  a  given  rate 

p  =  the  present  worth  of  $1  for  a  given  time  at  a  given  rate 

D  =  the  discount  on  $1  for  a  given  time  at  a  given  rate 

r    =  {i  -\-  i),  the  periodic  ratio  of  increase' 

Amount  of  Principal 

In  finding  the  amount  of  a  given  principal  at  compound  inter- 
est for  a  given  number  of  years  at  a  given  rate  per  year  com- 
pounded annually,  it  is  customary  to  compute  the  amount  of  $1 
for  the  given  time  at  the  given  rate,  and  multiply  this  result  by 
the  number  of  dollars  in  the  principal. 

When  i  =  6%  or  .06,  the  accumulation  of  the  amount  may  be 
computed  by  either  of  the  following  methods: 


-C.  E.  Sprague  and  L.  L.  Perrine,  The  Accountancy  of  Investment,  1914. 


l62         MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 

First  Method  Second  Method 

Dollars     End  of  Dollars         Symbols 

Principal i.oo  i.oo                    I 

Interest  on  Ji. 00 .06  Multiply  by  ...  .      1.06               (i   +  i) 

1.06              I  year  1.06              (i  +  «') 

"          "      1.06 0636  "           "    ....      1.06              (i  + »') 

1. 1236    2  "  1. 1236    (i  + «')' 

"    "  1. 1236 067416  "     "  ....   1.06      (i  + ») 


1.191016  3   "  1.191016   (i  +»)^ 

1.191016 071461  "     "  ....   1.06      (i  + ») 

1.262477  4   "  1.262477   (i  +')* 


The  ratio  of  increase  is,  in  symbols,  (i+i),  and  in  figures  1.06. 
The  investment  at  the  beginning  of  each  year  must,  therefore,  be 
multiplied  by  (i+i)  to  obtain  the  investment,  or  amount,  at  the 
end  of  that  year.  Hence  i  invested  at  the  rate  i  becomes  at  the 
end  of 

1  year (i  +  O 

2  " d  +  O' 

3  " (i-hi)' 

4  " d+O' 

n    " (i  +  O" 

Thus  the  formula  is  obtained 

a=  (1+  i)" 

If  it  is  desired  to  find  the  amount  of  i  at  5%  for  20  years, 
compounded  annually,  the  formula  becomes 

Frequency  of  Compounding 

In  the  preceding  illustration  the  interest  was  compounded 
annually.  But  the  period  of  compounding  may  be  shorter  than 
one  year,  with  the  result  that  the  compounding  occurs  with 
greater  frequency  than  once  a  year.  In  such  cases  the  formula 
stated  above  still  applies.    Although  compounding  may  occur 


SIMPLE  AND  COMPOUND  INTEREST  163 

semiannually,  quarterly,  monthly  or  even  daily,  the  rate  is 
usually  stated  as  a  certain  per  cent  per  year,  but  i  in  the  formula 
is  the  annual  rate  divided  by  the  number  representing  the  periods 
in  a  year.  For  instance,  if  the  rate  is  6%  per  year,  iunder  various 
conditions  would  be  as  follows: 

Frequency  of 

Compounding  Value  of  i 

Annually .06 

Semiannually (.06  -^  2  )  .03 

Quarterly (.06  -^  4  )  .015 

Monthly (.06  -^  12  )  .005 

Daily (.06  -^  365)     -^ 

The  number  of  periods,  represented  in  the  formula  by  n,  will 
be  the  number  of  years  times  the  number  of  periods  per  year. 
For  instance  n,  under  various  conditions,  is  as  follows: 

Frequency  of  Value  of  n 

Compounding        One  Year     Two  Years     Three  Years 

Annually i  2  3 

Semiannually.  .  .  2  4  6 

Quarterly 4  8  12 

Monthly 12  24  36 

Daily 365  730  1095 

Illustration 

What  is  the  amount  of  $1  at  6%  interest  for  20  years,  compounded 
quarterly? 

Solution: 

i  =  .06  -4-  4  =  .015,  the  rate  for  one  period 

n  =  20  X  4  =  80,  the  number  of  periods 

a  =  (i    +  i)**  or  (1.0x5)  *°,  which  amounts  to  3.29066279 

Determining  the  Amount 

Interest  tables  show  the  amount  of  $1  at  various  rates  for 
various  periods.     Such  a  table  appears  in  the  Appendix  of  this 


1 64 


MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 


book.  When  a  table  is  not  available,  the  amount  may  be 
computed  with  a  table  of  logarithms.  When  neither  an  inter- 
est table  nor  a  table  cf  logarithms  is  available,  it  is  necessary  to 
compute  the  amount  by  repeated  multiplication,  thus: 


1.03 
1-03 

1. 0609 
1.03 

1.092727 
1.03 


a  =   (1.03)'=   ? 

=   (1.03)^ 
=  (1.03)^ 


1.12550881  =   (1.03)" 

At  this  point  the  number  of  decimal  places  becomes  so  large  as 
to  make  the  computation  too  laborious.  The  number  may  be 
reduced  to  six  places,  by  approximation,  thus: 

1.12550881  becomes  1. 125509 

The  multiplication  continued  is  as  follows: 

1.125509  =  (i.03)'* 
1.03 


159274 
03 


(1-03) ■ 


194052  =   (1.03)' 
03 


229874  =   (1.03)7 
03 


266770  =   (1.03) 

This  last  amount  is  the  amount  of  $1  at  6%  interest 
for  4  years,  compounded  semiannually 


SIMPLE  AND  COMPOUND  INTEREST  165 

But  when  the  number  of  periods  is  large,  as  will  be  the  case 
when  the  interest  must  be  calculated  over  a  long  term  of  years, 
these  repeated  multiplications  become  irksome.  It  is  possible, 
however,  to  cut  down  on  the  number  of  operations  necessary.  A 
short  method  may  be  utilized  which  should  be  readily  understood 
if  it  is  remembered  that  (14-^)"  means  that  (i  +i)  is  used  n  times 
as  a  factor.  If  we  know  the  value  of  (i  +i)  ^  we  have  the  amoimt 
obtained  by  using  (i+i)  twice  as  a  factor.  By  squaring  this 
amount  we  get  (1+^)'',  which  is  the  product  resulting  from  the 
use  of  (1+/)  four  times  as  a  factor.  Squaring  this  amount  in 
turn  gives  us  (i  +i)  ^,  the  product  obtained  by  using  (i  +/)  eight 
times  as  a  factor. 

By  the  application  of  this  principle,  a  very  material  reduction 
may  be  effected  in  the  number  of  multiplications,  as  is  made  clear 
in  the  example  given  below : 


1.03 
1.03 

1.0609 
1.0609 

=  (i  +  i) 
=  (i  +  i) 

1. 125509 
1. 125509 

=  (1  +  i)' 

1.266770  =  (i  +  i) 

The  quantity   1.03'°  can   be  determined  in   the  following 
manner : 

1.26677     (i  +  «)^ 
1.0609      (i  "f"  ^)^ 


1. 34391 6  (i  +  i)^" 

The  general  application  of  this  principle  may  be  made  clearer 
by  another  illustration. 


I66         MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 

Illustration 

Required — the  value  of  (i  +  i)^°: 

Solution: 

(I  +  i) 
(I  +  i) 


:  + 

i)' 

+ 

iy 

+ 

i)' 

+ 

i)' 

+ 

i)' 

+ 

iV 

+ 

■yU 

+ 

i)' 

+ 

iy 

+ 

iy 

+ 

iV 

+ 

iy 

Thus,  only  seven  multiplications  are  required  to  obtain  the  30th 
power. 

The  80th  power  could  be  obtained  thus : 

(i  +  7)  '^  (obtained  by  four  multiplications) 

(1+  i)" 


(1+  i)'' 
(1  +  i)'' 

u  +  iy'' 


SIMPLE  AND  COMPOUND  INTEREST  167 

The  same  principle  may  be  utilized  in  the  use  of  compound 
interest  tables  which  do  not  extend  to  the  desired  number  of 
periods. 

Illustration 

Given  a  table  of  40  periods,  required — the  value  of  (i  +  i)  '^. 

Solution: 

( I  +  /)  '^ "  shown  by  table 

(I  +  /V     "      "     " 


(i  +  i)" 


Determining  Interest 

The  amount  of  $1  at  compound  interest  is  composed  of  two 
elements:  the  original  investment  of  $1  and  the  accumulated 
interest.  Hence,  to  find  the  compound  interest,  apply  the  follow- 
ing formula: 

/  =  c  —  I 
or  /  =   (i  +  J-)"  -  I 

If  $1.266770  is  the  amount  of  $1  at  6%  interest,  compounded 
semiannually  for  four  years,  $1 .000000  is  the  original  investment, 
and  $.266770  is  the  compound  interest. 

Determining  Present  Worth 

The  present  worth  of  a  sum  due  at  a  fixed  future  date  is  a 
smaller  sum  which  with  interest  will  amount  to  the  future  sum. 
For  instance,  the  present  value  of  $1  due  in  one  year  at  6%  is  a 
sum,  smaller  than  $1,  which  with  interest  at  6%  for  one  year  will 
amount  to  $1.     Representing  the  present  value  of  this  $1  by  p 

p  X  1.06  =  $1 

or  in  symbols,  pX  (i  +  i)  =  i  when  the  dollar  is 

due  in  one  year. 


I68         MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 

Since  pX  {i  +  i)  =  i, 

it  follows  that  i  -r-  (i  +  t)  =  /» 

(Since  p  X  1.06  =  i,  it  follows  that  I-^  1.06  =  .943396.) 
If  the  dollar  is  due  two  periods  hence, 

PX  (i-i-  i)'=  I 
and  it  follows  that  i  -^  (i  +  f)^  =  /> 

If  the  dollar  is  due  three  periods  hence 

pX  (i-\-  i)'=  I 
and  it  follows  that  i  -^  {i  -\-  i)^  =  p 

If  the  dollar  is  due  ?i  periods  hence 

px  {i-\-  ir  =  I 

and  it  follows  that  i  -^  (i  -\-  i)"  =  p 

But  since  (i  +  i)"  =  a 

the  formula  p  =  i  -^  (i  +  i)" 

may  also  be  stated  p  =  i  -^  a 

In  determining  the  present  value  at  6%  of  $1  due  in  four 
years,  interest  compounded  semiannually,  several  methods  are 
available. 

1 .  It  may  be  possible  to  refer  to  a  table  of  present  values, 
similar  to  the  one  in  the  Appendix. 

2.  If  a  table  of  present  values  is  not  available,  the  present 
value  may  be  computed  by  using  the  formula 

p  =  I  -T-  a 
It  will  first  be  necessary  to  obtain  the  value  of  a.  When  in- 
terest at  6%  per  annum  is  compounded  semiannually,  7  =  .03 ; 
with  semiannual  compounding  for  four  years,  «  =  8.  Then,  a  = 
(1.03)^.  The  value  of  a  may  be  found  by  any  of  the  methods 
previously  explained;  it  is  1.266770.     Then 

$1  -V-  1.266770  =  $.789409,  the  present  value 


SIMPLE  AND  COMPOUND  INTEREST 


169 


3.  Instead  of  dividing  $1  by  (1.03) ^  i.e.,  by  1.266770,  the 
same  result  can  be  obtained  by  dividing  by  1.03  eight  times,  using 
I  as  the  first  dividend,  each  succeeding  dividend  being  the  quo- 
tient resulting  from  the  preceding  division,  thus 


$1.000000  - 

-  1.03  =  $. 

.970874  - 

-  1.03  =  . 

.942596  - 

-  1.03  =  . 

.915142  - 

-  103  =  . 

.888487  - 

-  I-03  =  . 

.862609  - 

-  1.03  =  . 

.837484  - 

-  1.03  =  . 

.813092  - 

-  1.03  =  . 

970874,  present  value  of  $ 

942596, 

915142, 


862609, 

837484, 
813092, 
789409, 


due  in  i  period 

2  periods 

3 

4 

5 

6 

7 


Determining  the  Compound  Discount 

As  shown  in  the  preceding  section,  one  would,  in  exchange  for 
a  promise  to  pay  $1  a  given  number  of  periods  hence,  loan  the 
present  value  of  $1.  Present  values  for  various  numbers  of 
periods,  and  the  discount  earned,  are  shown  below  (discounted 
at  3%  per  period)  : 


Due  Periods 
Hence 


Amount  Loaned 
Present  Value 


I $.970874 

2 942596 

3 915142 

4 888487 

8 .789409 


Discount 

Earned 

(i-p) 

$.029126 
.057404 
.084858 

•111513 
.210591 


The  discount  for  two  or  more  periods  is  called  compound 
discount.    Compound  discount  may  be  computed  by  the  formula 

D  =  I  -  p 

It  may  also  be  computed  by  the  formula 

D=  I  -^  a 


170         MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 

This  formula  requires  explanation.  Compound  discount  is 
really  compound  interest,  deducted  in  advance;  but  it  is  the 
compound  interest  on  the  money  actually  loaned. 

For  instance,  if  $.78940915  loaned  on  a  promise  to  pay$i  eight 
periods  hence  at  3%,  the  discount  $.210591  is  3%  compound 
interest  on  $.789409,  the  principal  actually  loaned.  This  fact 
can  be  demonstrated  thus : 

The  amount  of  $1  at  3%  compound  interest  for  eight  periods 
is  $1.266770.  Hence  the  compound  interest  on  $1  is  $.266770 
and  the  compound  interest  on  $.789409  is  the  result  in  the  fol- 
lowing multiplication: 

.789409  p 

Multiplied  by  .266770  i 

Product  .210591  d 

Since  the  compound  discount  {D)  is  really  the  compound  interest 
(/)  on  the  actual  loan  {p) , 

D=  IX  p 
And  since  />  =  i  -^  a, 

we  can  substitute  (i  H-a)  for  p,  and  the  formula  becomes 

D=  IX  I 
or  D  =  I  -h  a 

This  formula  may  also  be  explained  thus :  The  present  value  of 
$1  due  in  eight  periods  at  3%  is  $.789409,  or  $1  -^(1.03)^,  or$i-^a. 
If  the  loan  were  i,  the  compound  interest  would  be  /,  or 
$.266770;  but  since  the  loan  is  i-i-a,  the  compound  discount  is  / 
-7-a,  or  $.266770^  1. 266770=$. 210590. 

Summary 

In  this  chapter  the  following  formulas  have  been  derived: 

a   =  (i  +  ir 

I  =  a  —  I 

p  =  1  -r-  a;        ori-T-(i  +  t)'' 

D  =  1  —  p,        or  /  -7-  a 


CHAPTER  XVII 

ANNUITIES 

Definition  of  Annuities 

A  series  of  equal  payments,  due  at  regular  intervals,  is  an  an- 
nuity. Although  the  word  "annuity"  suggests  annum  and  year, 
the  interval  may  be  any  period,  as  a  month,  quarter  or  half-year. 

Symbols 

In  the  discussion  of  annuities  the  following  symbols  will  be 
employed : 

A  =  the  amount  of  an  annuity  of  $i  for  a  given  time  at  a  given 

rate 
P  =  the  present  value  of  an  annuity  of  |i  for  a  given  time  at  a 

given  rate 

Amount  of  an  Annuity 

Let  us  assume  that  a  contract  provides  for  payments  as  in  the 
following : 

Example 

January  i,  191 5 $100 

"        I,  1916 100 

"        I,  1917 100 

"        I,  1918 100 

Total $400 

Required — the  accumulated  value  of  this  annuity  at  January  i,  1918, 
interest  at  5%. 

Solution:  It  is  assumed  that  each  payment  is  put  at  interest,  in 
which  case  the  amount  of  each  payment  computed  separately,  is  as  follows: 

171 


172 


MATHEMATICS  OP  ACCOUNTING  AND  FINANCE 


Payment  Made 


January  i,  19 is 

I,  1916 

I,  1917 

"         I,  1918 

Amount  of  the  annuity 


Periods  at 
Interest 


Payment 


$100 
100 
100 
100 


Amount 


Symbol 


(I  +i)' 
(I  +.)' 
(I   +  0 


115.7625 
110.2500 
105.0000 
100.0000 


431.0125 


Although  the  amount  of  an  annuity  may  be  computed  by 
determining  the  amount  of  each  payment,  it  is  unnecessary  to 
resort  to  this  labor,  as  the  following  short  method  may  be  used: 

To  find  the  amount  of  an  annuity  of  $i  for  a  given  number  of 
periods  at  a  given  rate,  divide  the  compound  interest  on  $i  for  the 
number  of  periods  at  the  given  rate,  by  the  interest  rate. 

Or,  in  symbols,  A  =  I  -^  i 

Applying  this  formula  to  the  illustration  above,  /,  the  compound 
interest  on  $i  for  four  periods  at  5%,  is  shown  by  an  interest  table 
to  be  $.215506; then 

$  .215506  -J-  .05  =  $4.31012,  amount  of  an  annuity  of  $1 
$4.31012   X  100  =  $431,012,  amount  of  an  annuity  of  $100 

This  formula  requires  explanation.  Let  us  suppose  that  $1  is 
loaned  on  January  i,  1914,  the  contract  requiring  payment  of 
simple  interest  annually  at  5%  as  follows: 

D.\TE  Interest 

January  i,  igi5 $.05 

I,  1916 .05 

I,  1917 05 

1,1918 05 

These  payments,  being  equal  in  amount  and  made  at  regular 
intervals,  constitute  an  annuity  of  $.05  per  year.  If  put  at  5% 
compound  interest,  the  amount  of  each  payment  computed 
separately  would  be  as  follows: 


ANNUITIES 


173 


Payment  Made 


January  i,  ipis 
I,  1916 
I.  1917 
I,  1918 

Totals 


Periods  at 
Interest 


Payment 


$.05 

•  OS 

•  OS 

•  OS 


f.20 


Amount 


Symbols 


(l  +  i)i 
(I  +i}' 
(I   +  i) 


.057881 
•055125 
.052500 
.050000 


.215506 


The  total  of  the  annuity  payments  is  $.20,  the  simple  interest 
on  $1  for  four  years ;  and  the  amount  of  the  annuity,  $.215  506,  is  the 
compound  interest  on  $1  for  four  years,  or  /.  Hence,  compound 
interest  is  merely  the  amount  of  an  annuity.  If  the  compound 
interest  $.215506  is  the  amount  at  5%  of  four  annual  payments  of 
5  cents, 

$.215506 -^  5  =  $.0431012,  the  amount  at  5%  of  four  annual  pay- 
ments of  $.01 


or         $.215506  -^  .05  = 


or  in  symbols 


I..31012,  the  amount  at  5%  of  four  annual 
payments  of  $1. 

1  ^  i=  A 


Illustration 

Required  the  amount  of  an  annuity  of  $25  per  month  for  five 
years  at  6%  per  annum,  interest  compounded  monthly. 


Solution: 

12  (payments  per  year)  X  5  =  60,  number  of  terms 
6%  -^  12  =  K%  or  -ooS)  the  rate 

Now,     I  =  a  —  1 

And      a  =  (1.005)^°=  1.348849 

Then     \  =  I  -^  i 

=  .348849  ^  .005 

=  69.7698,  amount  of  an  annuity  of  $1 
$69.7698  X  25  =  $1,744.25,  amount  of  an  annuity  of  $25 


174         MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 

Sinking  Fund  Contribution 

The  preceding  section  developed  a  method  of  determining 
the  amount  which  results  from  the  accumulation  of  a  known 
annuity.  It  is  the  purpose  of  this  section  to  discuss  the  converse 
problem,  of  finding  an  unknown  annuity  which  will  produce  a 
required  amount. 

This  problem  finds  application  among  accountants  in  com- 
puting the  periodical  contribution  necessary  to  accumulate  a 
required  sinking  fund.  Let  us  assume  that  a  sinking  fund  of 
$100,000  is  to  be  accumulated  in  five  years  by  equal  instalments 
made  at  the  end  of  each  year;  what  is  the  required  contribution, 
assuming  4%  interest  compounded  annually? 

The  annual  contributions  constitute  an  annuity  and  the 
accumulated  fund  is  the  amount  of  the  annuity.  We  shall  first 
find  what  fund  would  be  accumulated  by  a  contribution  of  $1, 
applying  the  formula 

A  =  I  ^  i 


To  find  I: 

1.04 
Multiplied  by  1.04 

1.0816 
"   1.0816 

=  (1  +  i)' 

1.16Q85856 
"            "   1.04 

=  (1+  /)^ 

1. 2166529024 
Deduct                                 I . 

=  (i  +  f)5 

.2166529024  =  /,  compound  interest  on  $1 
for  five  periods  at  4% 

To  find  A  : 

.2166529024  -^  .04  =  5.41632256,  amount  of  an  annuity  of  $1 


ANNUITIES 


175 


Since  annual  contributions  of  $1  will  produce  a  fund  of 
$5.41632256,  to  find  the  contributions  necessary  to  produce  a 
fund  of  $100,000: 

$100,000  -r-  5.41632256  =  $18,462.71,  required  contribution 

The  computation  was  performed  without  using  an  interest 
table.  If  tables  are  available,  the  work  may  be  materially 
decreased. 

For  instance,  a  table  of  amounts  of  $1  per  annum  shows  the 
amount  of  $1  for  live  periods  at  4%  to  be  $5.416323.  Only  one 
computation  is  necessary : 

$100,000  -j-  5.416323  =  $18,462.71 

If  only  a  compound  interest  table  is  available,  it  will  show 
the  amount  of  $1  in  five  periods  at  4%  to  be  $1.216653.  The 
following  computations  will  be  necessary: 

$.216653 -^     -04  =  $5.416325,  amount  of  annuity  of  $1 

$100,000  -^  5.416325  =  $18,462.71 

The  following  tabulation  of  sinking  fund  accumulations  shows 
the  accumulation  of  the  fund  from  the  two  elements,  annual 
contributions  and  interest: 


End  of  Ye.-vr 

Contribution 

Interest 

Total  Fund 

$18,462.71 
18,462.71 
18,462.71 
18,462.71 
18,462.71 

$    738.51 
1,506.56 
2,305.33 
3.136.05 

$18,462.71 
37.663.93 
57.633.20 
78,401.24 

Totals 

$92,313.55 

$7,686.45 

Present  Worth  of  an  Annuity 

Let  us  assume  that  a  contract,  made  on  January  i,  1914, 
provides  for  the  following  payments : 


176 


MATHEMATICS  OP  ACCOUNTING  AND  FINANCE 


January  i,  1915 $100 

"        I,  1916 100 

"        I,  1917 100 

"        I,  1918 100 

Total   $400 


We  are  to  find  the  present  value  of  this  series  of  payments, 
or  annuity,  on  January  i,  1914,  discounted  at  5%. 

The  present  value  of  each  payment,  computed  separately,  is 
as  follows: 


Payment  Due 

Periods  Hence 

Payment 

Present  Value 

Symbol 

$ 

I 

2 
3 
4 

lioo 
100 
100 
100 

I  -=-  Cr   4-  i) 

95.2381 
90.7029 

86.3838 

82.2702 

I,  1916  . 

I  - 
I  - 

-  (I  +i)' 

-  (i  +i)^ 

-  ri  4-  .-14 

I,  1917 

I,  1918 

Totals 

J400 

3S4.S9S0 

The  present  value  of  each  payment,  and  the  present  value  of 
the  annuity,  may  be  computed  thus: 


5100 
95.2381 
90.7029 
86.3838 


-T-    1.05  =   $  95.2381     present  value  of  $100  due  in    i    year 


1.05  =  90.7029, 
1.05  =  86.3838, 
1.05  =        82.2702, 


$100 
$100 
$100 


2  years 

3 

4 


$354.5950,     present  value  of  the  annuity. 


When  interest  tables  are  not  available,  this  method  is  satis- 
factory; but  w^ith  interest  tables  available  a  short  method  may  be 
used  which  is  similar  to  the  short  method  of  computing  the 
amount  of  an  annuity. 

To  find  the  present  value  of  an  annuity  of  $1  for  a  given 


ANNUITIES 


177 


number  of  periods  at  a  given  rate,  divide  the  compound  discount 
on  $1  for  the  given  number  of  periods  by  the  interest  rate.  Or 
in  symbols, 

P  =  D  -^  t 

Applying  this  formula  to  the  illustration  above,  the  present  value 
of  $1  due  four  periods  hence  at  5%,  is  shown  by  an  interest  table 
to  be  $.822702; 

D,  the  compound  discount,  is  $.177298. 

$.177298  -4-  .05  =   $3.54596,  the  present  value  of  an  annuity  of  $1 

$3-54595  X  100  =  $354,596,    "         "  "       "    "         "         "  $100 

This  formula  also  requires  explanation.  Let  us  suppose  that 
a  loan  of  $1  is  made  on  January  i,  1914,  the  contract  requiring 
the  payment  of  simple  interest  annually  at  5%  as  follows : 

Date  Interest 

January  i,  191 5 $.05 

"         I,  1916 05 

I,  1917 05 

"         I,  1918 05 

These  payments,  being  equal  in  amount,  and  made  at  regular 
intervals,  constitute  an  annuity  of  five  cents  per  year.  If  dis- 
counted at  5%,  the  present  value  of  each  payment  on  January 
I,  1 9 14,  computed  separately,  is  as  follows: 


Payment  Due 

Periods  Henxe 

Payment 

Present  Value 

Symbols 

$ 

January  i,  1915 

I,  1916 

I 
2 
3 

4 

S.os 

•  05 

•  OS 
■  05 

I  - 
I  - 
I  - 

-  (1  +  .) 

-  (i  +.)' 

-  (i  +.-)3 

-  Ct    -I-  ,U 

.0476191 
•0453514 
.0431919 
.0411351 

Present  value  of  the  annuity  .... 

•177297s 

178         MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 

Now,  instead  of  paying  five  cents  interest  each  year,  and  pay- 
ing the  $1  at  maturity,  the  present  value  ($.1772975)  of  the  four 
interest  payments  might  be  paid,  or  deducted,  in  advance,  thus: 

$1.0000000  payment  to  be  made  at  end  of  four  years 
.1772975  present  value  of  interest  payments 


$  .8227025  present  value  of  $1 


In  other  words,  the  simple  interest  on  $1  at  5%  for  four  years  is  an 
annuity  of  five  cents;  and  $.  1772975,  the  present  value  of  these  in- 
terest payments,  is  the  present  value  at  5%  of  an  annuity  of  four 
five  cents  payments;  and$.i  772975  is  also  the  compound  discount 
on  $1  due  four  periods  hence  at  5%.  Therefore  the  compound 
discount  on  $1 ,  due  in  four  periods  at  5%,  is  the  present  value  of 
an  annuity  of  five  cents  in  four  periods  at  5%. 

If  the  compound  discount  $.1772975  is  the  present  value  at 
5%  of  four  annual  payments  of  5  cents,  then 

$.1772975  H-  5  =  $.0354595,  the  present  value  at  5%  of  four  annual 

payments  of  i  cent 

or    $.1772975  -4-  .05  =  $3.54595,  the  present  value  at  5%  of  four 

annual  payments  of  $1 

or  in  symbols  D  -^  i  =  P 

Illustration 

Required  the  present  value  of  an  annuity  of  $25  per  month  for  five 
years  at  6%  per  annum,  compounded  monthly. 

Solution:  12  (payments  per  year)  X   5  =   60,  number  of  periods 
6%  -i-  12  =  K%  or  .005,  the  rate 

D  =   1  -  p 
and     />  =   I  -^  (1.005)^"=  .741372 

Then  D  =  i  —  .741372  =  .258628,  the  compound  discount  on  $1 

due  in  sixty  periods  at  >2  % 


ANNUITIES  179 

and 

D  ^  i=  P 

$     .258628  -^  .005  =  $        51.7256,  present  value  of  an  annuity 

of  $1 
$51.7256       X     25  =  $1,293.14,        the  present  value  of  an  an- 
nuity of  $25 

Rent  of  an  Annuity 

The  preceding  section,  containing  an  explanation  of  the 
method  of  determining  the  present  worth  of  a  known  annuity, 
developed  the  formula,  P  =  D-^i.  Each  periodical  instalment  of 
an  annuity  is  known  as  the  rent  of  an  annuity.  This  section  will 
deal  with  the  question  of  determining  what  rent  will  be  produced 
by  a  known  present  worth. 

It  has  been  shown  that  the  present  value  of  an  annuity  of  $1 
for  four  periods  at  5%  is  $3.54595, 

Since  a  present  value  of  $3.54595  will  produce  four  annual 
rents  of  $1   at  5%,  a  present  value  of  $1   will  produce  four 

annual  rents  of of  $1,  or  $.282012. 

3-54595 

Therefore,  to  find  the  rent  of  an  annuity  with  a  present  value 
of  $1  for  a  given  number  of  periods  at  a  given  rate: 

1.  Find  the  present  value  of  an  annuity  of  $1  for  the  given 

time  and  rate 

2.  Divide  $1  by  this  present  value  of  an  annuity  of  $1 

Or  expressed  in  symbols, 

R=  I  -^  P 

Illustration 

A  man  invests  $3,000  in  an  annuity  to  be  repaid  to  him  in  five  annual 
instalments,  interest  computed  at  6%  annually.  What  annual  rent  will 
be  produced? 


l8o         MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 

Solution:  Formula — R  =  i  -^  P 

The  present  value  of  $i  due  five  periods  hence  at  6%  may  be  com- 
puted thus: 

I  -=-  (i.o6)s 

or  it  may  be  determined  from  interest  tables.     The  compound  interest 
table  shov/s 

1. 06^    =  1.338226 
then 

I  -^    1.338226  =    .747258,  the  present  value  of  $1  due 

in  five  periods 

I  —     .747258  =    .252742,  the  compound  discount  on  $1 

due  in  five  periods 
.252742     -T-  .06  =  4.21236,  the  present  value  of  an  annuity 

of  $1  for  five  periods  at  6% 

then 

R  =  1  -7-  4.21236  =  .237396,  rent  produced  by  present  worth  of 

$1 
$.237396  X  3,000  =  $712.19,  rent  produced  by  present  worth  of 

$3,000 

The  following  schedule  shows  the  reduction  of  the  investment 
due  to  the  excess  of  the  periodical  payments  over  the  interest 
earned  on  the  decreasing  investment. 

Original  investment $3,000.00 

Interest  ist  year,  6%  of  $3,000 180.00 

$3,180.00 
Deduct  ist  rent 712.19 

Balance,  end  of  ist  year $2,467.81 

Interest  2d  year,  6%  of  $2,467.81 148.07 

$2,615.88 
Deduct  2d  rent 712.19 

Balance,  end  of  2d  year $1,903.69 

Interest  3d  year,  6%  of  $1,903.69 114.22 

$2,OI7.Qi 

Deduct  3d  rent 712.19 


ANNUITIES  l8l 

Balance,  end  of  3d  year $1,305.72 

Interest  4th  year,  6%  of  $1,305.72 78.34 

$1,384.06 
Deduct  4th  rent 712.19 

Balance,  end  of  4th  year $671.87 

Interest,  5th  year,  6%  of  $671.87 40.31 

$712.18 
Deduct  5th  rent 712.18 

$       .00 

The  schedule  could  also  be  arranged  as  follows,  showing  that 
each  payment  is  composed  of  two  parts: 

1 .  Interest  on  the  diminishing  principal 

2.  Repayment  of  the  principal 

Original  investment $3,000.00 

First  rent: 

Interest  on  $3,000  at  6% $180.00 

Repayment  of  principal 532.19  532.19 

$712.19     $2,467.81 

Second  rent: 

Interest  on  $2,467.81  at  6% $148.07 

Repayment  of  principal 564.12  564.12 

$712.19     $1,903.69 

Third  rent: 

Interest  on  $1,903.69  at  6% $114.22 

Repayment  of  principal 597-97  597-97 

$712.19     $1,305.72 

Fourth  rent: 

Interest  on  $1,305.72  at  6% $  78.34 

Repayment  of  principal 633.85  633.85 

$712.19     $    671.87 
Fifth  rent: 

Interest  on  $671.87  at  6% $  40.31 

Repayment  of  principal 671.87  671.87 

$712.18     $  .00 


l82         MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 
Or  the  schedule  may  be  shown  thus: 
Payment  Rent 

I $    712.19 

2 712.19 

3 712.19 

4 712.19 

5 712.18 

Totals $3,560.94 


[ntere-st 

Reduction  of 
Investment 

Diminishing 
Investment 

$3,000.00 

$180.00 

$ 

532.19 

2,467.81 

148.07 

564.12 

1,903.69 

114.22 

78.34 

40.31 

$3 

597-97 
633-85 
671.87 

,000.00 

1,305-72 

671.87 

0 

$560.94 

Equal  Periodical  Payments  on  Principal  and  Interest 

When  a  debt  together  with  the  interest  thereon  is  to  be  paid  in 
equal  periodical  instalments,  the  principal  of  the  debt  is  the  pres- 
ent value  of  an  annuity,  and  the  periodical  payments  are  rents, 
to  be  computed  by  the  method  already  explained. 

To  illustrate,  if  $3,000  bearing  6%  interest  compounded 
annually,  is  to  be  paid  in  five  annual  instalments,  each  to  include 
the  accrued  interest  and  a  portion  of  the  principal,  $3,000  is  the 
present  value  of  five  unknown  rents.  These  rents  were  computed 
in  the  preceding  illustration ,  being  $712.19.  The  reduction  of  the 
debt  may  be  tabulated  thus : 

Original  principal $3,000.00 

First  payment $712.19 

Less  I  year's  interest  on  $3,000 180.00 

Payment  on  principal 532.19 

Balance  of  principal $2,467.81 

Second  payment $712.19 

Less  I  year's  interest  on  $2,467.81  ....        148.07 

Payment  on  principal 564.12 

Balance  of  principal $1,903.69 

Third  payment $712.19 

Less  I  year's  interest  on  $1,903.69 114.22 

Payment  on  principal 597-97 


ANNUITIES  183 

Balance  of  principal $1,305.72 

Fourth  payment $712.19 

Less  I  year's  interest  on  $1,305.72 78.34 

Payment  on  principal 633,85 


Balance  of  principal $671.87 

Fifth  payment $712.18 

Less  I  year's  interest  on  $671.87 40.31 

Payment  on  principal 671.87 

Balance  of  principal $      .00 


Annuities  Due 

In  the  foregoing  discussion  of  annuities,  rents,  and  sinking 
fund  contributions,  the  formulas  and  methods  described  apply  to 
the  ordinary  form  of  annuities,  in  which  the  payments  are  made 
at  the  end  of  the  periods. 

When  the  payments  are  made  at  the  beginning  of  the  periods, 
the  annuity  is  called  an  annuity  due.  Changing  the  payment 
from  the  end  to  the  beginning  of  the  period  affects  the  compound 
interest,  thus  changing  both  the  amount  and  the  present  value  of 
the  annuity. 

To  Find  the  Amount  of  an  Annuity  Due 

To  arrive  at  a  method  of  computing  the  amount  of  an  annuity 
due,  let  us  compare  the  amounts  of: 

1.  An  ordinary  annuity  (A)  of  six  periods  (due  at  end  of 

period) 

2.  An  annuity  due  (B)  of  live  periods  (due  at  beginning  of 

period) 

In  each  annuity,  the  interest  is  compoimded  annually  at  4%. 
A  requires  the  payment  of  $1  on  December  31  for  each  of  six 
years,  beginning  December  31,  191 1. 


1 84 


MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 


B  requires  the  payment  of  $i  on  January  i  for  each  of  five  years, 

beginning  January  i,  191 2. 
It  will  be  noted  that  the  first  five  payments  of  annuities  A  and  B 
are  made  on  practically  identical  dates;  but  annuity  A  has  one 
more  payment  than  annuity  B ;  hence  the  amount  of  annuity  A 
will  exceed  the  amount  of  annuity  B  by  one  payment  of  $1. 
This  may  be  more  clearly  shown  by  the  following  table  which 
shows  the  amount  of  each  annuity  payment  and  the  amount  of 
the  annuity. 


Annuity  A 

Annuity  B 

Date 

Payment 

Amount  at 
Dec.  31.  1916 

Date 

Payment 

Amount  at 
Dec.  31.  1916 

Dec.  31,  1911 

•'     31.  1912 

"    31.  1913 

"    31.  1914 

"    31.  191S 

"    31,  1916 

Ji.oo 
1. 00 
1. 00 
1. 00 
1. 00 
1. 00 

Si. 216653 
1. 169859 
1. 124864 
1. 081600 
1.040000 
1. 00 

Jan.  I,  1912 

"     I.  1913 

"     I.  1914 

"     I.  191S 

"     I,  1916 

$1.00 
1. 00 
1. 00 
1. 00 
r.oo 

Si. 216653 
1. 169859 
1. 1 24864 
1. 081600 
1.040000 

$6.632976 

55-632976 

Annuity  B  is  an  annuity  due  of  five  rents;  annuity  A  is  an  or- 
dinary annuity  of  six  rents;  the  periodical  payments  and  interest 
rate  are  the  same  in  each  case.  The  difference  between  the  two 
amounts  is  $1,  or  one  periodical  payment. 

Hence  we  can  find  the  amount  of  annuity  B  (annuity  due)  for 
five  periods  by  ascertaining  the  amount  of  an  ordinary  annuity 
(A)  of  the  same  payment  and  rate  for  six  periods,  and  deducting 
$1 .   The  amount  of  annuity  A  (ordinary)  may  be  computed  thus: 

$1.265319,  amount  of  $1  for  six  periods 
1. 00 


$  .265319,  compound  interest  for  six  periods 


$.265319  -^  .04  =  $6.632975,  amount  of  an  ordinary  annuity  of  six  periods 


ANNUITIES  185 

To  compute  the  amount  of  B,  the  annuity  due: 

$6.632975,  amount  of  ordinary  annuity  of  six  periods 
Deduct       1. 000000,  one  rent 

$5.632975,  amount  of  annuity  due  of  five  periods 

Sinking  Fund 

Contributions  to  the  sinking  fund  of  a  bond  issue  are  ordin- 
arily made  at  the  end  of  the  period.  If  a  bond  issue  is  to  run 
twenty  years  and  provision  is  to  be  made  for  it  by  twenty  con- 
tributions to  a  sinking  fund,  the  first  contribution  is  usually  made 
at  the  end  of  the  first  year,  thus  allowing  time  in  which  to  acquire 
cash  from  profits.  Such  contributions  to  a  fund  constitute  an 
ordinary  annuity. 

When  contributions  to  a  fund  are  made  at  the  beginning  of  the 
period,  the  payments  become  an  annuity  due.  To  find  the 
periodical  contribution  necessary  to  accumulate  the  required 
fund  by  such  payments : 

1.  Find  the  amount  of  an  ordinary  annuity  of  $1  for  one 

more  than  the  number  of  periods 

2.  Deduct  $1  in  order  to  find  the  amount  of  an  annuity 

due  of  $1  for  the  given  number  of  periods 

3.  Divide  the  required  fund  by  the  amount  of  an  annuity  due 

of  $1. 

Required  Annual  Contribution 

In  the  discussion  of  sinking  fund  contributions  in  the  early 
part  of  this  chapter  it  was  required  to  find  the  sinking  fund 
contribution  necessary  to  invest  at  4%  at  the  end  of  each  of  five 
years  to  pay  an  obligation  of  $100,000.  This  was  found  to  be 
$18,462.71.  We  shall  now  calculate  the  required  annual  contri- 
bution if  made  at  the  beginning  of  each  year. 

These  payments  constitute  an  annuity  due  of  five  periods. 
The  calculation  will  proceed  by  the  following  steps: 


1 86 


MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 


I.  Find  the  amount  of  an  ordinary  annuity  of  six  periods: 
!i. 265319,  amount  of  $1  at  compound  interest  for  six  periods  at  4% 


.265319,  compound  interest  on  $1  for  six  periods  at  4% 

.265319  -r-  .04  =  $6.632975,  amount  of  ordinary  annuity  of  six  periods 

2.  Find  the  amount  of  an  annuity  due  of  five  periods: 

$6.632975 


.632975,  amount  of  annuity  due  of  five  periods 


3.  Divide  total  fund  by  amount  of  annuity  of  $1 : 

$100,000  -H  5.632975  =  $17,752.61,  required  annual  contribution 

The  following  sinking  fund  table  shows  the  accumulation  of 
the  fund  due  to  the  two  elements  of  contributions  and  compound 
interest. 


Ye.\r 

Contribution 

Fund  Beginning  Year 

Interest 

Fund  End  of  Ye.\r 

I 
2 
3 
4 
5 

$17,752.61 
17,752.61 
17,752.61 
17,752.61 
17,752.61 

517.752-61 
36,215.32 
55.416.54 
75,385.81 
96,153.85 

$      710.10 
1,448.61 
2.216.66 
3.015.43 
3.846.15 

$   18,462.71 

37,663.93 

57,633.20 

78,401.24 

100,000.00 

$88,763.05 

Six, 236. 95 

Present  Worth  of  an  Annuity  Due 

When  the  annuity  payments  are  due  at  the  beginning  of  the 
period,  the  present  value  of  the  annuity  is  composed  of  two 
elements : 

1.  A  rent,  or  payment,  now  due  (a) 

2.  The  present  value  of  an  ordinary  annuity  of  one  less  than 

the  given  number  of  periods  (b) 


ANNUITIES  187 

For  instance,  $5.451822  deposited  at  4%  interest  compounded 
annually  permits  the  immediate  withdrawal  of: 

(a)  $1  rent  now  due,  and  leaves  a  balance  of 

(b)  $4.451822,  which  is  the  present  value  of  live  annual  pay- 

ments of  $1,  the  first  of  which  is  due  one  year  hence. 

Therefore,  to  find  the  present  value  of  an  annuity  due,  proceed  as  in 
the  following  illustration. 

Illustration 

What  is  the  present  value  at  4%  per  annum  of  an  annuity  of  six  pay- 
ments of  $1 ,  the  first  of  which  is  due? 

Solution:     i.    Find  the  present  value  of  an  ordinary  annuity  for  one 
less  than  the  given  number  of  periods: 

$1.000000 

.821927,  present  value  of  $1  due  in  five  years  at  4% 

$  .178073,  compound  discount  on  $1  due  in  five  years  at  4% 
$  .178073  -^  .04  =  $4.451825,  present  value  of   an  ordinary  annuity  of 
five  payments 

2.   To  this  present  value  add  one  rent: 

$4.451825,  present  value  of  ordinary  annuity  of  five  payments 
1. 00 

$5.451825,  present  value  of  an  annuity  due  of  six  payments 

The  reduction  of  this  present  value,  due  to  the  accumulation 
of  interest  and  the  payment  of  rents  is,  tabulated  below. 

Present  value  of  annuity  due  of  six  rents $5.451825 

First  rent i . 

Balance $4.451825 

Interest  earned  ist  period .178073 

Balance  end  of       "     "        $4.629898 

Second  rent i . 


I88         MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 

Balance  beginning  2d  period $3.629898 

Interest  earned       "       "        .145196 

Balance  end  of        "       "         $3.775094 

Third  rent i. 

Balance  beginning  3d  period $2.775094 

Interest  earned          "       "       .111004 

Balance  end  of          "     "         $2.886098 

Fourth  rent i. 


Balance  beginning  4th  period $1. 

Interest  earned  "       "         .075444 

Balance  end  of  "       "        $1.961542 

Fifth  rent i . 

Balance  beginning  5th  period $  .961542 

Interest  earned  "       "         .038461 

Balance  end  of  "       "         $1.000003 

Sixth  rent i . 

Since  the  sixth  rent  is  withdrawn  at  the  beginning  of  the  sixth 
period,  there  is  no  balance  at  the  beginning  of,  nor  interest  earned 
during,  the  sixth  period. 

Rents 

The  converse  of  the  above  problem  is  to  determine  what 
periodical  payment  or  rent  a  known  present  value  will  produce, 
when  the  first  payment  is  due  immediately. 

For  instance,  what  rent  will  an  investment  of  $1,000  produce, 
the  first  of  six  annual  rents  being  due  at  once  and  the  interest 
being  4%? 

1.  Find  what  present  value  will  produce  a  rent  of  $1 

This  was  found  above  to  be  $5.451825 

2.  Divide  the  known  present  values  by  the  present  value  of 

an  annuity  of  $1 

$1000.00  -j-  5.451825  =  $183.42 


ANNUITIES  189 

The  reduction  of  this  present  value  may  be  tabulated  thus: 


Period 

Rent 

Balance 

Interest 

Balance 

$1,000.00 

I 

3183.42 

$816.58 

I32.66 

849.24 

2 

183.42 

665.82 

26.63 

692.4s 

3 

183.42 

sog.03 

20.36 

529.39 

4 

183.42 

345.97 

13.84 

3S9.8I 

S 

183.42 

176.39 

7.06 

183.4s 

6 

183.4s 

0. 

Si. 100.55 

$100.55 

CHAPTER  XVIIi 

LOGARITHMS  IN  COMPOUND  INTEREST  AND 
ANNUITY   COMPUTATIONS 

Calculating  Compound  Interest  and  Annuities  by  Logarithms 

The  following  examples  illustrate  the  methods  of  utilizing 
logarithms  in  compound  interest  and  annuity  computations. 

Examples 

I.   To  find  the  amount  of  i.     Required — the  amount  of  $4,500  at  4% 
per  annum  for  twenty  years,  compounded  semiannually 

Solution: 

Amount  =  $4,500  X  1.02''° 

Log  1.02  =  .00860 

Multiply  by  40 


Log  1.02''"      =  .34400,  the  log  of  2.208 

Hence  1.02''''=  2.208 

Amount  =  $4500  X  2.208  =  $9936 

Amount  computed  with  aid  of  interest  table  =  $9936.18 

Logarithms  may  be  used  more  extensively  in  this  solution,  as  follows: 

Log  4,500  =     365321 

Log  1.02  =  .00860 
Multiply  by        40 

Log  1.02''"  =      .34400 

Log  4,500+  log  1.02'"'=    3.99721,  log  of  9,936 
Hence  $4,500  X  1.02'"'=  $9,936 

2.  To  find  the  compound  interest  on  i. 

Solution:  Compute  the  amount  as  above  and  deduct  the  principal 

Amount  $9,936. 

Principal  4.500. 

Compound  interest    $5,436. 
iqo 


LOGARITHMS  AND  ANNUITY  COMPUTATIONS 


191 


3.  To  find  the  principal.     What  sum  invested  at  4%  compounded 
quarterly  will  amount  to  $5000  in  eight  years? 
Solution: 

Principal  =  5000  -7-  (i.oi)-*^ 

Log  5000  =     3.69897 

Log  1. 01        =  .00432 
Multiply  by  32 


Log  i.oi^^  =      .13824 


Log  of  principal  =    3.56073,  the  log  of  3636.92 

Principal  =  $3636.92 

Principal  computed  by  interest  table  =  $3636.52. 

4.  To  find  the  rate.     If  $1,125  is  to  be  returned  at  the  expiration  of 
seven  years  for  a  loan  of  $800,  what  rate  of  interest  compounded  an- 
nually is  earned? 
Solution: 

Since    1,125     =  800  X  (i  +  tV 

(i  +  i)^=     1,125  -J-  800 
Log  1,125         =  305115 


Log  800 

Log  (i +1)7 
Log  I  +  i 
Hence  \  -\-  i 

i 


90309 


14806 

14806  -7-  7  =  .021151,  the  log  of  1.0499 

0499 

0499,  or  nearly  5% 


5.   To  find  the  time.     For  how  many  years  should  $2,000  be  placed  at 
5%  interest  compounded  annually  to  produce  $5,054? 

Solution: 

Since  $5,054  =  $2,000  X  1.05" 
5,054 -^    2,000  =  1.05" 
Hence  3.70364,  which  is  log  5054 
Minus  3.30103,      "      "    "    2,000 


Equals  .40261         ''      "     "    1.05" 
Since  .40261  =  log  1.05  X  n 

n  =  .40261  -T-  log  1.05 
Since  log  1.05  =  .02119 

n  =  .40261  -^  .02119  —  19)  the  number  of  years 


192         MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 

6.  To  find  the  present  value  of  i.     What  is  the  present  value  of  $15,000 
due  in  five  years  at  4%,  interest  compounded  quarterly? 

Solution: 

Present  value  =  $15,000 -^  (i.oi)^" 

Log  15,000  =      4.17609 

Log  1. 01  =  .00432 

Multiply  by  20 

Log  i.oi^"  =         .08640 

Log  of  present  value  =       4.08969 

Present  value  =  $12,293.89 

Present  value  by  compound  interest  table  =  $12,293.17 

7.  To  find  the  compound  discount  on  i. 

Solution:  Compute  the  present  value  as  in  the  preceding  example: 
deduct  this  present  value  from  the  amount  due  at  maturity. 
Amount  due  at  maturity  $15,000.00 

Present  value  12,293.89 

Compound  discount  $  2,706.11 

8.  To  find  the  amount  of  an  annuity.     What    is    the    amount    of    an 
ordinary  annuity  of  $25  for  twenty  periods  at  4%  per  period? 

Solution: 

1.04'"-  I 

Amount  of  an  annuity  of  $1  = 

.04 

Log  1.04  =     .01703 

Multiply  by  20 


Log  1.04^"  =    .34060,  the  log  of  2.1908 

1.04^"  =  2.1908 

Deduct  1. 0000 

Compound  interest  =  1.1908 

Amount  of  annuity  of  $  i  =  $  1.1908  ^  .04  =  $  29.77 
"       "  "       "    $25  =     29.77      X  25  =  $744-25 

"       "         "       "      "   by  interest  tables  =  $744-45 
Logarithms  may  be  used  more  extensively  in  these  computations,  but 
it  is  desired  to  make  the  solutions  as  simple  as  possible. 

9.  To  find  the  amount  of  sinkiui^  fund  contributions.  What  annual 
contribution  must  be  made  at  the  end  of  each  of  twenty  years  to  amount 
to  $50,000  at  4K%? 


LOGARITHMS  AND  ANNUITY  COMPUTATIONS  I93 

Solution: 

$50,000  -r-  amount  of  annuity  of  $1  =  contribution 

Log  1.045                 .                             =  .01912 

Multiply  by  20 

Log  1.045^"  =     .38240,  the  log  of  2.41 21 1 1 

Hence  1.045^°  =  2.41 21 11 

Deduct  1. 000000 


1.412111 


Compound  interest 

Amount  of  annuity  of  $1  =  1.412111  -^  .045  =  31.38 

$50,000-=-  31.38  =$1593-37 

10.   To  find  the  present  worth  of  an  annuity.     What  is  the  present  value 
of  an  annuity  of  $50  per  year  for  ten  years  at  6%? 

Solution:  The  present  value  of  the  annuity  =  50  X  I> 

D  =.-         ^ 


1.06 

Log  1.06         =  .02531 
Multiply  by  10 


Log  of  1. 06'°  =  •-?53io,  the  k-gof  amount  uf  i  .00 

Determine  the  present  value  as  follows: 

Log  I.  =  0.00000  or  10.00000  —  10 

Log  1.06'"  =  .25310 


Hence  log——  =  g. 74690  -  10,  the  log  of  .5583375 


1.06 
1. 0000000 
Minus     .5583375 


Equals    .4416625,  the  compound  discount 

.4416625  -^  .06  =     7.36104,  present  value  of  annuity  of  $  i 
$7.36104      X    50  =  $368,052,       "  "       "         "        "  $50 


CHAPTER  XIX 
BOND   DISCOUNT   AND   PREMIUM 

Bonds  Purchased  below  and  above  Par 

Bonds  are  frequently  purchased  at  prices  either  below  or 
above  par;  that  is,  at  a  discount  or  a  premium.  When  a  bond 
purchased  at  a  discount  is  held  until  maturity  and  paid  at  par, 
the  owner  makes  a  profit  amounting  to  the  difference  between  the 
cost  and  par;  that  is,  to  the  discount.  If  the  bond  is  purchased 
at  a  premium  and  repaid  at  par,  the  owner  incurs  a  loss  amount- 
ing to  the  premium.  Conversely,  one  who  issues  a  bond  below 
par  and  repays  it  at  par  loses  the  discount,  while  one  who  issues 
a  bond  at  a  premium  and  repays  it  at  par  gains  the  premium. 
The  question  arises  as  to  when  such  profit  shall  be  credited,  or 
such  loss  be  charged,  to  profit  and  loss.  The  following  discussion 
considers  the  subject  from  the  point  of  view  of  the  investor;  the 
same  principles  apply  to  writing  off  premium  or  discount  on  the 
books  of  the  business  issuing  the  bonds. 

Discount 

Let  us  assume  that  a  $ioo  bond,  due  in  two  years  and  bearing 
4%  interest  payable  semiannually,  is  bought  for  $96.28.  When 
paid  at  par  there  will  be  a  gain  of  $3.72,  the  amount  of  the  dis- 
count. The  discount  may  be  taken  up  by  the  following  methods, 
which  are,  however,  unscientific: 

I.  By  an  immediatecredit  to  income  of  $3.72,  thus  raising  the 
investment  account  on  the  books  to  the  par  of  $100  and  taking 
credit  at  once  for  the  discount.  This  method  is  clearly  wrong 
because  it  takes  up  at  the  time  of  purchase  an  income  which  is 
earned  gradually  as  the  bond  approaches  maturity. 

194 


BOND  DISCOUNT  AND  PREMIUM 


195 


2.  By  carrying  the  investment  at  its  cost  of  $96.28  until  it  is 
paid,  at  which  time  the  difference  between  the  cash  received  and 
the  cost  of  the  bond  is  credited  to  income.  This  method  also  is 
erroneous,  although  the  error  is  not  so  apparent.  In  the  first 
place,  while  the  earning  is  made  gradually  as  the  bond  approaches 
maturity,  it  would  appear  from  the  above  treatment  that  none 
of  the  discount  was  earned  until  the  very  day  of  maturity;  and  in 
the  second  place  the  bond  increases  in  value  as  the  day  of  matur- 
ity approaches,  when  it  is  paid  at  par. 

3.  By  considering  the  $3.72  as  extra  interest  earned,  divid- 
ing the  amount  by  4,  the  number  of  semiannual  interest  periods, 
charging  }i  of  $3.72  each  six  months  to  the  bond  account  and 
crediting  it  to  interest.  This  has  the  effect  of  raising  the  invest- 
ment account  gradually  to  par,  while  spreading  the  earning  over 
the  four  periods  in  which  the  bond  is  held.  The  effect  of  this 
treatment  is  shown  by  the  following  schedule: 


Period 

Bond  account 

beginning  of 

6  months  period 

Portion  of  discou.nt 

TAKEN   UP   during   ONE 
PERIOD 

Int.  collected 

Total 
credit  to 
interest 

I 
2 
3 

4 
Maturity 

$    96.28 
97-21 
98.14 
99.07 
100.00 

%    .93 
.93 
.93 
■93 

$2.00 

2.00 

2.00 
2.00 

J    2.93 
2.93 
2.93 
2.93 

Total 

I3.72 

$8.00 

$1  1.72 

It  may  appear  that  this  method  answers  all  requirements, 
since  it  raises  the  investment  account  gradually  to  par,  taking 
up  the  discount  periodically.  But  it  is  subject  to  the  criticism 
that,  while  the  investment  is  gradually  increasing,  the  interest 
remains  constant.  This  means  that  the  rate  of  interest  is  gradu- 
ally decreasing,  as  shown  in  the  following  table: 


196 


MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 


Period 

Asset  value  in- 
bond  ACCOUNT 

Credit  to  Income 

Semiannual 
rate  of  income 

I 
2 
3 

4 

$96.28 
97-21 
98.14 
99-07 

f2.93 
2.93 
2.93 
2.93 

3.043  +% 

3-014   +  % 

2.98s  +  % 

2.957   +  % 

Since  the  investment  or  asset  value  is  increasing  as  the  bond 
approaches  maturity,  the  credits  to  income  should  also  increase, 
so  that  the  same  rate  of  income  will  be  maintained  each  period. 
When  the  amounts  are  small,  the  error  arising  from  this  method 
is  immaterial  and  the  discount  may  be  written  off  in  equal 
amounts,  as  above.  But  when  the  amounts  are  large,  this 
method  may  result  in  serious  unfairness  to  some  parties  in 
interest.  Assuming,  for  instance,  that  $96,280  is  invested  in 
the  purchase  of  $100,000  of  bonds  for  a  trust  and  that  the  bene- 
ficiary changes  at  the  expiration  of  the  second  six  months'  period, 
the  first  beneficiary  would  receive  over  3%  income  semiannually 
on  the  assets  of  the  trust,  while  the  second  beneficiary  would 
receive  less  than  3%.  It  is  not  sufficient  that  they  each  receive 
the  same  number  of  dollars.  The  property,  or  investment,  held 
for  the  second  beneficiary  is  of  greater  value  than  that  held  for 
the  first,  and  hence  the  second  should  receive  more  income  per 
period,  since  each  is  entitled  to  the  same  rate  of  income  on  the 
investment  of  the  trust. 


Scientific  Method  of  Amortization 

Any  method  of  writing  off  discount,  to  be  scientific,  must  take 
the  following  into  consideration : 

1.  The  investment  increases  in  value  as  the  bond  approaches 
maturity,  and  at  each  interest  date  the  investment  value  should 
be  increased  on  the  books,  thus  raising  it  gradually  to  par. 

2.  The  purchase  of  a  bond  at  a  discount  results  in  the  rate  of 
interest  earned  on  the  investment  being  higher  than  the  nominal 


BOND  DISCOUNT  AND  PREMIUM  197 

rate  paid  on  the  par  of  the  bond.  Hence,  the  amount  added  to 
the  asset  value  of  the  bond  at  each  interest  date  should  also  be 
credited  to  interest,  together  with  the  cash  collected. 

3.  The  total  credit  to  interest  (cash  collected  plus  portion  of 
discount)  at  the  end  of  each  period  should  be  an  increasing 
amount  but  always  the  same  per  cent  of  the  carrying  or  asset 
value  of  the  bond  at  the  beginning  of  the  period. 

When  a  bond  is  bought  at  a  price  other  than  par,  there  are  two 
interest  rates: 

1 .  The  nominal  or  cash  rate  paid  on  the  par  of  the  bond 

2.  The  effective,  basic,  or  income  rate  actually  earned  on  the 

investment 

When  the  bond  is  bought  at  a  discount,  the  income  rate  is  greater 
than  the  nominal  rate  for  two  reasons : 

1.  The  investment  is  less  than  par,  although  it  gradually 

increases  to  par. 

2.  The  income  is  more  than  the  coupons  collected,  since  the 

income  is  composed  of  two  parts : 

(a)  The  coupons  collected 

(b)  The  periodical  portion  of  the  discount 

When  the  discount  is  scientifically  amortized  or  written  off  the 
method  is  as  follows: 

1 .  Determine  the  effective  rate  earned  on  the  investment,  as 

explained  later. 

2.  At  each  interest  date  multiply  the  carrying  value  of  the 

investment  by  the  effective  rate,   to  determine  the 
amount  of  revenue  earned  during  the  period. 

3.  Make  the  following  entry: 

Debit  cash  for  coupon  collected 

Debit  investment  for  portion  of  discount 

Credit  interest  for  total  income  computed  in  (2)  above. 


198         MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 

The  amount  charged  to  investment  will  be  the  difference 
between  the  total  income  and  the  cash  collected. 

To  illustrate,  the  price  of  $96.28  in  the  example  above,  was 
chosen  because  it  was  known  that  a  4%  bond  due  in  two  years, 
with  semiannual  interest,  would  yield  an  effective  rate  of  income 
of  6%  per  year  or  3%  semiannually  if  purchased  at  this  price. 
The  following  shows  the  scientific  amortization  of  the  discount : 

Cost  of  bond $96.28 

First  period: 

Income,  3%  of  $96.28 $2.89 

Coupon,  2%  of  $100.00 2.00 

Balance — portion    of    discount    charged    to 

investment .89 

Second  period: 

Carrying  value  of  bond $97- 1 7 

Income,  3%  of  $97.17 $2.92 

Coupon 2.00 

Balance — portion  of  discount  charged  to  in- 
vestment    -92 

Third  period: 

Carrying  value  of  bond $98.09 

Income,  3%  of  $98.09 $2.94 

Coupon 2.00 

Balance — portion    of    discount    charged    to 

investment -94 

Fourth  period: 

Carrying  value  of  bond $99-03 

Income,  3%  of  $99-03 $2.97 

Coupon 2.00 

Balance — portion  of  discount  charged  to  in- 
vestment    -97 

Par  of  bond $100.00 


BOND  DISCOUNT  AND  PREMIUM 


199 


It  will  be  noted  that  this  process  conforms  to  the  three  require- 
ments of  the  scientific  method  as  outlined  above. 

The  investment  is  increased  gradually  to  par  by  writing  off  a 
portion  of  the  discount  at  each  interest  period,  thus: 


Beginning  of 

Period 

Investment 

I 

$96.28 

2 

97.17 

3 

98.09 

4 

Q9-03 

Par 

100.00 

The  amount  added  to  the  investment  value  at  each  interest 
period  is  also  credited  to  interest,  as  follows: 

Credits  to  Interest 


End  of  Period 

For  C.\sh 

For  Portion'  of  Discount 

TOT.\L 

I 
2 
3 

4 

$2.00 
2.00 
2.00 
2.00 

$   .89 
.92 
•94 
•  97 

$2.89 
2.92 
2.94 
2.97 

The  total  credit  to  interest  increases  each  period  and  is  always 
the  same  per  cent  of  the  increased  investment  at  the  beginning  of 
each  period,  as  shown  below: 


Period 

Investment,  beginning  of  period 

'Income  for  period 

R.\TE 

I 
3 
3 
4 

$96.28 
97.17 
98.09 
99-03 

$2.89 
2.92 
2.94 
2.97 

•i  /o 
3% 
3% 
3% 

200 


MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 


While,  as  stated  before,  the  difference  is  immaterial  when  the 
investment  is  small,  the  following  table  comparing  results  by  the 
equal  instalment  method  and  the  scientific  amortization  method 
will  serve  to  show  the  injustice  which  might  be  caused  by  the 
improper  method. 


Investment  Value 

Income 

Equal 

Scientific 

Differ- 

Equal 

Scientific 

Differ- 

INSTAL. PLAN 

AMORTZN.  PLAN 

ence 

INSTAL.  PLAN 

AMORTZN.  plan 

ence 

I 

$96.28 

I96.28 

i.oo 

52. 93 

$2.89 

S+-04 

2 

97-21 

97-17 

.04 

2.93 

2.92 

+.01 

3 

98.14 

98.09 

-05 

2-93 

2.94 

— -01 

4 

99-07 

99-03 

-04 

2.93 

2.97 

-.04 

Assuming  that  the  bond  was  held  as  an  investment  for  a  trust 
and  that  the  beneficiary  changed  at  the  end  of  the  second  period, 
the  column  of  income  differences  shows  that,  by  the  equal  instal- 
ment method,  the  first  beneficiary  would  receive  5  cents  too 
much  income,  while  the  second  beneficiary  would  receive  5  cents 
less  than  was  rightfully  his. 

Assuming  that  the  bond  was  held  for  an  estate  for  the  benefit 
of  a  life  tenant  and  a  remainderman,  that  the  life  tenant  died  at 
the  end  of  the  second  half  year,  and  that  the  estate  reverting  to 
the  remainderman,  the  equal  instalment  method  would  work 
detriment  to  the  remainderman.  Let  us  assume  that  the  estate 
originally  consisted  of  $100.  By  the  equal  instalment  method 
the  estate  reverting  to  the  remainderman  would  be  less  than 
$100. 

The  income  represented  by  the  portion  of  discount  written 
off  would  be  paid  to  the  life  tenant  in  cash  and  would  be  added  to 
the  carrying  value  of  the  bond. 

The  following  would  be  a  statement  of  the  assets  of  the  estate: 


BOND  DISCOUNT  AND  PREMIUM 


20I 


Equal  Instalment 

Method 

1 

Scientific  Amortiza- 
tion Method 

Cash 

Bond 
true  value 

Total 

Cash 

Bond 
TRUE  value 

Total 

$100.00 

96.28 
$     3.72 

2.00 

J96.28 
97-17 
98.09 

$100.00 
100.00 

9996 

99-95 

$100.00 

96.28 

J96.28 
97.17 
98.09 

First  period: 

Cash  collected  ... 

$     3-72 
2.00 

$100.00 

$     5-72 
2.93 

$     5.72 
2.89 

$      2.79 
2.00 

S     2.83 
2.00 

100.00 

Second  period: 
Cash  collected  .  . 

S     4-79 
2.93 

$     4-83 
2.92 

S      1.86 

S     1-91 

100.00 

It  is  seen  that  if  the  equal  instalment  method  was  followed, 
the  assets  of  the  estate  would  be  impaired  5  cents  by  paying  the 
life  tenant  in  cash  5  cents  erroneously  computed  as  income.  The 
result  would  be  that  the  remainderman  would  be  defrauded, 
the  injustice  being  concealed  by  turning  the  bond  back  to  him 
at  a  value  of  $98.14  when  it  really  had  a  value  of  only  $98.09. 


Income  Rates 

Bond  dealers  frequently  offer  bonds  at  prices  which  net  the 
investor  a  rate  other  than  the  cash  or  nominal  rate.  In  the 
illustration  above,  where  a  4%  bond  was  purchased  to  yield  an 
income  rate  of  6%,  the  quotation  might  be  stated,  "4%  bond 
paying  6%,"  "4%  bond  yielding  6%,"  "4%  bond  to  net  6%,"  or 
' '  4%  bond  on  a  6%  basis. ' '  To  avoid  the  necessity  of  computing 
the  price  at  which  each  sale  must  be  made,  bond  tables  have 


202         MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 


This  price  varies 


been  prepared  showing  the  price  to  be  paid, 
with 

1 .  The  nominal  rate 

2.  The  income  rate 

3.  The  number  of  periods  mi  til  maturity 

Since  bonds  usually  bear  semiannual  interest  there  are  two 
periods  per  year,  and  bond  tables  usually  show  a  price  based  on 
the  assumption  that  the  interest  is  paid  semiannually. 

The  following  portion  of  a  page  in  a  bond  table  shows  various 
prices  to  be  paid  for  a  2-year  bond,  depending  on  the  nominal 
and  effective  rates.  The  rates  at  the  head  of  the  columns  are 
nominal  rates;  those  at  the  side  are  the  effective  rates.  For 
instance,  in  the  4%  nominal  rate  column  on  the  6%  effective  rate 
line  is  found  the  price  of  $96.28  used  in  the  preceding  illustrations 
of  a  2-year  4%  bond  netting  6%. 


2  Years 

Interest  Payable  Semiannually 

Per  Cent 
Per  Annum 

3% 

3  1/2% 

4% 

4  1/2% 

5% 

6% 

7% 

4.80 

96.61 

97.55 

98.49 

99.43 

100.38 

102.26 

104.IS 

4  7/8 

96.47 

97.41 

98.3s 

99.29 

100.24 

102.12 

104.00 

4.90 

96.42 

97.36 

98-31 

99-25 

100.19 

102.07 

103.95 

S 

96.24 

97.18 

98.12 

99  06 

100.00 

101.88 

103.76 

5-10 

96.05 

96.99 

97-93 

98.87 

99.81 

101.69 

103.57 

S  1/8 

96.01 

96.95 

97.89 

98.83 

99.77 

101.64 

103.52 

S-20 

95.87 

96.81 

97-75 

98-69 

99.62 

101.50 

103.38 

5  1/4 

95.78 

96.72 

97-66 

98.59 

99-53 

101.41 

103.28 

5.30 

95.69 

96.63 

97-56 

98.50 

99-44 

101.31 

103.19 

5  3/8 

95-55 

96.49 

97.43 

98.36 

99-30 

101.17 

103.04 

S-40 

95.51 

96.44 

97.38 

98.32 

99-25 

I0I.I2 

103.00 

SI/2 

95-33 

96.26 

97-20 

98.13 

99-07 

100.93 

102.80 

5  5/8 

95-10 

96.03 

96.97 

97.90 

98.83 

100.70 

102-57 

53/4 

94-87 

95.81 

96.74 

97-67 

98.60 

100.47 

102.33 

5  7/8 

94-65 

95-58 

96.51 

97-44 

98.37 

100.23 

102.09 

6 

94.42 

95-35 

96.28 

97-21 

98.14 

100.00 

101.86 

BOND  DISCOUNT  AND  PREMIUM  203 

Bond  Premium 

When  a  bond  is  bought  at  a  price  above  par,  the  effective  rate 
is  less  than  the  nominal  rate,  for  two  reasons: 

1 .  The  investment  is  more  than  the  par  to  which  the  nominal 

rate  applies. 

2.  The  coupons  collected  are  not  all  income;  since  only  par 

will  be  repaid,  a  portion  of  the  cash  received  at  each 
interest  date  must  be  considered  as  a  return  of  the 
premium  (which  is  principal)  and  only  the  balance  as 
income. 

Bond  premium  should  be  scientifically  written  off  as  follows: 

1 .  Determine  the  effective  rate. 

2.  Multiply  the  gradually  diminishing  investment  by  the 

effective  rate  to  compute  the  income  earned  during  the 
period. 

3.  Make  the  following  entry: 
Debit  cash  for  coupon  collected 

Credit  income  for  amount  computed  in  (2)  above 

' '      investment  for  difference  between  cash  and  income. 

To  illustrate,  the  bond  table  shows  that  a  2-year  $100  6%  bond 
to  net  4%  should  be  purchased  for  $103.81 .  The  following  shows 
the  scientific  amortization  of  the  premium: 

Cost  of  bond $103.81 

First  period: 

Coupon,  3%  of  $100.00 $3.00 

Income,  2%  of  $103.81 2.08 

Balance,  premium  written  off .92 

Second  period: 

Carrying  value  of  bond $102.89 

Coupon $3.00 

Income,  2%  of  $102.89 2.06 

Balance,  premium  written  off .94 


204 


MATHEMATICS  OP  ACCOUNTING  AND  FINANCE 

Third  period: 

Carrying  value  of  bond $101.95 

Coupon $3.00 

Income,  2%  of  $101.95 2.04 

Balance,  premium  written  off .96 

Fourth  period: 

Carrying  value  of  bond $100.99 

Coupon $3.00 

Income,  2%  of  $100.99 2.02 

Balance,  premium  written  off .98 

Par  of  bond $100.01 


The  error  of  i  cent  arises  from  repeated  approximations  of 
interest  to  the  nearest  cent. 

The  following  table  shows  the  periodical  entries : 


Period 

Debit  Cash 

Credit  Interest 

Credit  Investment 
(or  Premium) 

Investment 

Cost 

$  3.00 
300 
300 
3- 00 

$2.08 
2.06 
2.04 
2.01* 

1  .92 

.94 
.96 
.99* 

$103.81 

I 
2 
3 

4 

102.89 
101.95 
101.99 
100.00 

Total 

$12.00 

18. 19 

$3.81 

'  Adjustment  of  i  cent  to  correct  discrepancy. 


The  second  column  shows  the  total  cash  collected;  the  third, 
the  amount  of  cash  taken  as  income;  the  fourth,  the  portion  of  the 
coupon  applied  to  repayment  of  the  premium  ($3.81)  which,  with 
the  final  repayment  of  the  bond,  completely  realizes  the  invest- 
ment. Since  a  portion  of  the  investment  is  realized  periodically, 
the  investment  gradually  diminishes  in  value.  Hence  the  credits 
to  interest  decrease  with  each  successive  period. 


BOND  DISCOUNT  AND  PREMIUM  205 

Computing  the  Premium  and  the  Price 

While  the  price  above  par  to  be  paid  for  a  bond  to  yield  an 
effective  rate  less  than  the  nominal  rate  may  be  found  in  a  bond 
table,  such  tables  are  not  always  available.  If  a  table  of  com- 
pound interest,  present  values  or  annuities  is  at  hand,  it  will 
be  of  assistance,  but  the  price  may  be  computed  without  tables 
of  any  kind.     Two  methods  are  explained,  as  follows: 

First  Method 

An  interest  bearing  bond  comprises  two  promises,  as  follows: 

1.  To   pay    the   par   at  maturity — in   the  last  preceding 

illustration,  $ioo  at  the  expiration  of  four  six-months 
periods. 

2.  To  pay  a  stipulated  amount  of  interest  periodically — in 

the  illustration,  $3  at  the  end  of  each  of  four  six-months 
periods. 

The  price  to  be  paid  for  the  bond  is  the  present  value,  discounted 
at  the  effective  rate,  of  all  cash  payments  promised.  In  the 
illustration,  the  price  is  the  sum  of 

1.  The  present  value  of  $100  due  four  periods  hence,  dis- 

counted at  2%  per  period. 

2.  The  present  value  of  an  annuity  of  $3  for  four  periods, 

discounted  at  2%  per  period. 

These  present  values  may  be  found  in  the  interest  table;  thus  the 
table  shows  that: 

Present  value  of  $1  due  four  periods  hence 
at  2%  =  $.923845 

$.923845  X  100  =  $92.3845,  P.  V.  of  principal 
Present  value  of  an  annuity  of  $1  for  four 
periods  at  2%  =  $3.807729 

$3.807729  X  3  =       11.4232,  P.  V.  of  coupons 


Total $103.8077 

or $103.81,  price 


206         MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 

If  the  table  does  not  show  present  values  but  does  show  com- 
pound interest,  the  computation  may  be  made  as  follows: 

The  table  shows  the  amount  of  $i  at  compound  interest  for 
four  periods  at  2%  to  be  $1.082432. 

Then, $1.00 -J-  1.082432=  $       .923845,  P.  V.of$i 

$.923845  X  100  =  $  92.3845,  P.  V.  of  principal 

Also,  $1.00  —   .923845  =  $       .076155, 
compound  discount,  four  periods 
$.076155  -^  .02  =  $    3.8077, 

P.  V.  of  annuity  of  $1  for  four  periods 
$3.8077  X3  =     1 1. 423 1,  P.  V.  of  coupons 

Total $103.8076,      price 


When  no  interest  table  of  any  kind  is  available,  recourse  may 
be  had  to  the  methods  explained  in  the  chapter  on  compound 
interest  and  annuities.  Perhaps  the  easiest  method  would  be 
to  compute  the  amount  of  $1  for  four  periods  at  2%  thus: 

1.02 
1.02 

1.0404 
1.0404 

1.082432 
The  procedure  following  would  be  as  shown  above. 

Second  Method 

This  method  determines  only  the  premium  to  be  paid  to 
reduce  the  income  from  the  nominal  rate  to  the  effective  rate. 
The  premium  so  determined,  added  to  the  par  of  the  bond,  com- 
prises the  total  price.  The  method  is  based  on  the  following 
reasoning : 

Let  us  assume  that  a  4%  annual,  or  2%  semiannual,  income 
is  required.  If  the  bond  bore  4%  interest,  the  price  would  be  par. 
Hence  the  payment  of  $100,  or  par,  entitles  the  holder  of  the 
bond  to  receive: 


BOND  DISCOUNT  AND  PREMIUM  20/ 

1.  Principal,  at  maturity 

2.  Interest,  $2  at  the  end  of  each  six  months 

But  if,  as  in  the  preceding  illustration,  the  bond  pays  3%  semi- 
annually, each  semiannual  coupon  collected  will  be  $3.  Of 
this,  the  payment  of  par  entitles  the  holder  to  receive  $2. 
And  the  payment  of  a  premium  entitles  him  to  receive  the  remain- 
ing $1.  Hence,  the  premium  is  the  sum  which  must  be  paid 
to  entitle  the  holder  to  collect  that  portion  of  the  periodical 
coupon  which  is  in  excess  of  the  product  obtained  by  multiplying 
the  par  of  the  bond  by  the  effective  periodical  rate. 

This  excess  interest  is  an  annuity.  In  the  case  of  a  2-year  6% 
bond  bought  to  net  4%,  the  annuity  is  $1  for  four  periods.  The 
premium  is  the  present  value  of  this  annuity,  discounted  at  the 
effective  rate,  and  may  be  computed  as  follows: 

$100  X  3%  (semiannual  cash  rate)         =  $3.00  coupon 

$100  X  2%  (         "  effective  rate)  =    2.00  effective  income  on  par 

Excess     $1.00 

The  present  value  at  2%  of  an  annuity  of  $1  for  four  periods  is 
shown  by  an  interest  table  to  be  $3.807729,  to  which  is  added 
the  par,  $100,  the  total  being  the  price  of  the  bond  or  $103.81. 

When  a  table  showing  the  present  value  of  an  annuity  is  not  at 
hand,  one  of  the  methods  already  explained  may  be  used  to  find 
the  present  value.  Perhaps  the  easiest  method  is  by  successive 
divisions,  as  follows: 

Example 

What  is  the  premium  to  be  paid  on  a  3-year  7%  bond,  interest  payable 
semiannually,  bought  to  net  5%;  par  $100,000? 

Solution: 

$100,000  X  3K%  (cash  rate  per  period)  =$3,500 

$100,000  X  2>^%  (effective  rate  per  period)     =    2,500 

Excess    $1,000 


208 


MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 


The  required  premium,  therefore,  is  the  present  value  at  2^%  of  an 
annuity  of  $1,000  for  six  periods. 


975.610  P.  V.  of  $1000  due  I  period  hence 


Now, 

$1,000.00  - 

-  1.025  = 

$ 

975.610 

975.610- 

-  1.025  = 

951.814 

951.814  - 

-  1.025  = 

928.599 

928.599  - 

-  1-025  = 

905-950 

905-950  - 

-  1.025  = 

883.854 

883.854  - 

-  1-025  = 

862.297 

$1000 
$1000 
$1000 
$1000 
$1000 


2  periods 

3  " 

4  " 
5 

6 


$     5,508.124,  premium 
100,000.000,  par 


$105,508.12,    price 

The  premium  could  also  be  computed  as  follows: 
Find  amount  of  $1  for  six  periods  at  2^%,  as  follows: 


1.025 
1.025 


1.050625  amount  of  $1  for  2  periods 
1.050625 


1.103813         "       "      I     "  4      " 
1.050625 

1.159693        "       "      I     "  6      " 

Find  compound  discount  on  $1  due  six  periods  hence  at  2^  % 
thus: 

$1  -^     1. 159693  =  $.862297  P.  V.  of  $1  due  six  periods  hence 
$1  —  $  .862297  =     .137703  compound  discount 
or  $.159693  (comp.  int.)  -^  1. 159693  (amt.)  =  $. 137703  compound  discount 

Find  present  value  of  annuity  of  $1 ,000  for  six  periods  at  2%  %, 
thus: 

%  -137703  -^   -025  =  $5.50812   P.  V.  of  annuity  of  $1 
$5.50812    X  1000  =  $5,508.12  P.  V.  of  annuity  of  $1,000 


BOND  DISCOUNT  AND  PREMIUM 


209 


The  following  table  shows  the  periodical  entries  and  the 
diminishing  balance  of  the  investment. 


Period 

Debit  Cash 

Credit  Interest 
2j^%  OF  Investment 

Credit 

Investment 

Investment 

Cost 

$  3,500 
3.500 
3.500 
3.500 
3,500 
3,500 

$  2,637.70 
2,616.15 
2,59405 
2,571-40 
2,548-19 
2,524-38 

$    862.30 
883.85 
905.95 
928.60 
951-81 
975-62 

•?I05,508.I3 
104.645.83 
103,761.98 
102,856.03 
101.927.43 
100,975-62 
100,000.00 

6 

Total 

|2I,000 

115,491.87 

55,508.13 

Computing  the  Discount  and  the  Price 

When  bond  tables  cannot  be  consulted  to  determine  the  price 
to  pay  for  a  bond  to  net  an  income  rate  higher  than  the  nominal 
rate,  the  price  may  be  computed  by  methods  similar  to  those 
described  for  determining  a  premium. 

First  Method.  The  first  method  consists  of  the  following 
three  steps: 

(a)  Compute  the  present  value  of  the  par,  discounted  at 

the  effective  rate. 

(b)  Compute  the  present  value  at  the  effective  rate  of  all 

coupons  to  be  collected. 

(c)  Add  the  foregoing  two  items,  the  sum  being  the  price  to 

be  paid. 

In  the  case  of  the  $100  2 -year  4%  bond  bought  to  net  6%,  at  a 
price  of  $96.28,  the  first  item  (a)  is  the  present  value  of  $100 
due  four  periods  hence  at  3%  (the  effective  rate)  per  period. 
The  second  item  (b)  is  the  present  value  of  an  annuity  of  $2  for 
four  periods  discounted  at  3%. 

These  present  values  may  be  found  in  a  book  of  tables,  thus : 


210         MATHEMATICS  OP  ACCOUNTING  AND  FINANCE 

Present  value  of  par: 

P.  V.  of  $1  due  in  four  periods  at  3%  is  $  .888487 

$.888487  X  100  =  $88.8487 

Present  value  of  coupons: 

P.  V.  of  annuity  of  $1  for  four  periods  at  3%  is  $3.717098 
$3.717098X2=  7-4342 

Total  price  $96.2829 

or  as  shown  by  the  bond  table     $96.28 

If  an  interest  table  is  not  available,  the  methods  already 
described  for  determining  present  values  may  be  used. 

2.  Second  Method.  This  method  determines  the  discount 
to  be  deducted  from  par.  Since  the  cash  rate  does  not  produce 
the  required  income,  the  seller  permits  the  deduction  of  an 
amount  which,  invested  at  the  effective  rate,  will  produce  the 
extra  periodical  income  required.  The  discount  is  the  present 
value  of  an  annuity  of  the  extra  income. 

For  instance,  in  the  case  of  the  $100  2-year  4%  bond  bought 
to  net  6%,  at  a  price  of  $96.28,  the  3%  effective  rate  is  equivalent 
to  $3  per  period,  while  the  2%  coupon  produces  only  $2  per 
period,  the  required  excess  being  $1  per  period  for  four  periods. 
The  present  value  of  3%  (the  effective  rate)  of  an  annuity  of  $1 
for  four  periods  is  shown  by  an  interest  table  to  be  $3.717098. 

Then  $100.00  par 
Less         3.72  discount 


$96.28  price 

When  neither  a  bond  table  nor  an  interest  table  can  be  used, 
one  must  resort  to  the  previously  described  method  of  comput- 
ing the  present  value  of  the  annuity. 

Illustration 

What  are  the  discount  and  the  purchase  price  of  a  3-year  5%  bond, 
interest  payable  semiannually,  bought  to  net  6%;  par  $100,000? 


BOND  DISCOUNT  AND  PREMIUM 


211 


Solution: 

$100,000  X  3%      (effective  rate  per  period) 
$100,000  X  23^%  (cash  "       "         "    ) 


)3,ooo 
2,500 

5    500 


Deficient  interest  per  period 

The  required  discount,   therefore,  is  the  present  value  at  3%  of  an 
annuity  of  $500  per  period. 

--  $485,437  P.  V.  of  $500  due  I  period 


Now,     $500 

-  I. 

Hence,    485-437  " 

-  i.( 

47I.2Q8- 

-  i.< 

457-571- 

-  I. 

444.244- 

-  I. 

431.305  - 

-  I. 

Discount 

03  =  471 

03  =  457 

03  =  444 

03  =  431 

03  =  418 


298 
571 
244 

305 
742 


500 
500 
500 
500 
500 


2  periods 

3 

4        " 

5 
6 


$2,708,597 


Then  par 
Less  discount 

Price 


$100,000.00 
2,708.60 

$  97,291.40 


The  discount  can  also  be  computed  as  follows: 

Find  the  amount  of  $1  for  six  periods  at  3%,  thus: 


1.03 
1.03 


1.0609      amount  of  $1  for  2  periods 
1.0609 


1. 125509        "       ••      I     "  4 
1.0609 

1. 194052        "       "      I    "  6 

Find  the  compound  discount  on$i  due  six  periods  hence  at  3%,  thus: 
$1.00 -^  1. 194052  =  $.837484  P.  V.  of  $1  due  six  periods  hence 
$1.00—     .837484=  $.162516  compound  discount 

Find  the  present  value  of  an  annuity  of  $500  for  six  periods  at  3% 


thus: 


$.162516  (comp.  dis.)  -J-  .03  =  $5.4172  P.  V.  of  annuity  of  $1 

$5.4172  X  500  =  $2708.60  present  value  of  annuity  of  $500,  or  discount 


212 


MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 


The  following  table,  or  schedule  of  amortization,  shows  the  periodical 
entries  and  the  remaining  balance  of  the  investment. 


Period 

Credit  Income 
3%  OF  Investment 

Debit 
Cash 

Debit 
Investment 

Investment 

I 

2 

3 
4 
5 
6 

$  2,918.74 
2,931-31 
2,944-24 
2,957-57 
2,971-30 
2,985-44 

1  2,500 
2,500 
2,500 
2,500 
2.500 
2,500 

$    418.74 
431-31 
444-24 

457-57 
471-30 
485-44 

$  97.291-40 
97.710.14 
98,141.45 
98,585.69 
99.043.26 
99.514-56 
100,000.00 

$17,708.60 

$15,000.00 

$2,708.60 

Purchases  at  Intermediate  Date 

In  the  preceding  explanations  of  methods  for  computing 
prices  for  bonds  either  at  a  premium  or  at  a  discount,  it  has  been 
assumed  that  the  purchase  occurred  on  an  interest  date.  When 
this  is  not  the  case,  the  customary  method  of  determining  the 
price  is  as  follows : 

1.  Compute  the  price  as  if  the  purchase  had  been  made  at  the 
next  preceding  interest  date;  also  the  price  as  if  the  purchase  were 
to  be  made  at  the  next  succeeding  interest  date.  The  difference 
is  the  portion  of  premium  or  discount  to  be  amortized  during  the 
period. 

2.  Such  a  proportion  of  this  premium  or  discount  is  amortized 
as  the  elapsed  time  between  the  preceding  interest  date  and  the 
date  of  purchase  bears  to  the  total  interest  period. 

3.  To  the  amortized  value  thus  obtained  add  the  accrued 
interest  at  the  nominal  rate. 

For  instance,  referring  to  the  table  on  page  209  let  us  assume 
that  the  interest  dates  are  January  i  and  July  i ;  and  that 
$105,508.13  is  the  value  on  Jan.  i,  1918,  on  a  5%  basis 

104,645-83  "   "     "     "  July  i>  1918,  "  "    "      " 

$        862.30  "     "   premium  to  be  amortized  during  the  period 


BOND  DISCOUNT  AND  PREMIUM  213 

If  the  purchase  is  made  on  February  i,  19 18,  one-sixth  of  the 
interest  period  has  elapsed,  hence  one-sixth  of  $862.30  should  be 
amortized,  thus: 

Value  at  January  i,  1918 $105,508.13 

Deduct  Ye  of  $862.30 i43-72 

$105,364.41 
Add  accrued  interest:  ^/g  of  $3500 583.33 

Price $105,947.74 

The  price  may  also  be  computed  by  adding  to  the  price  on 
January  i  the  accrued  interest  at  the  effective  rate,  thus: 

Price  on  January  i,  1918 $105,508.13 

Effective  interest  for  six  months: 

2H%  of  $105,508.13  =   $2,637.70 

Ve  of  $2,637.70 439-6i 

Price  on  February  i,  1918 $105,947.74 

Of  this  amount  $105,364.41  is  charged  to  investment  and  $583.33 
to  accrued  interest.  When  the  coupon  is  collected  on  July  i, 
it  is  applied  as  follows: 

Cash $3500.00 

Accrued  interest $  583.33 

Interest  earned 2,198.09 

Investment  amortization  ($862.30  —  $143.72).  ..  718.58 

The  investment  of  $105,364.41,  reduced  by  the  amortization  of 
$718.58,  is  novi^  carried  at  the  true  value  (seepage  209)  on  July  i, 
$104,645.83. 

While  this  is  the  customary  method  of  computing  a  flat  price, 
it  is  unfair  to  the  buyer,  since  he  advances  to  the  seller  $583.33 
accrued  interest  five  months  before  it  is  due. 

When  the  bond  is  to  be  purchased  at  a  discount,  the  propor- 
tion of  discount  to  be  amortized  for  the  fractional  period  should 
be  added.  Referring  to  page  212,  let  us  assume  that  interest  is 
payable  on  January  i  and  July  i,  that  $97,291.40  is  the  value  at 


214         MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 

January  i,  1918,  on  a  6%  basis,  and  that  the  transfer  is  to  be 
made  on  May  i,  1918. 

Then,    $97,710.14  is  the  value  on  July         i,  igi8 
97,291.40  "    "        "       "  January  i,  1918 

$      418.74   "      "  discount   to   be  amortized   during   the 
period 
%  of  $418.74  =  $  279.16  the  discount  to  be  amortized  during  two-thirds 
of  the  period 

$97,291.40  value  at  Jan.  i,  1918 

279.16  two-thirds  of  discount  amortized  in  six  months 
1,666.67  accrued  interest — two-thirds  of  $2,500 

$99,237.23  price  on  May  i,  1918 

Or,  value  at  January  i,  1918 $97,291.40 

Add  two-thirds  of  3%  (effective  rate)  on  $97,291.40         1,945.83 

$99,237.23 

Of  this  amount  $97,570.56  is  charged  to  investment  and 
$1,666.67  to  accrued  interest.  On  July  i,  the  investment  is 
adjusted  to  its  true  value  (see  table  on  page  212)  by  the  following 
entries : 

Cash $2500.00 

Investment  (discount  $418.74  —  $279.16) 139-58 

Accrued  interest $1666.67 

Interest 972.91 

All  entries  after  July  i,  1918,  are  as  indicated  in  the  schedule 
on  page  212. 

Serial  Bonds 

Instead  of  providing  a  sinking  fund  for  the  eventual 
redemption  of  a  bond  issue,  the  bonds  may  be  retired  gradually 
by  serial  redemption.  In  computing  the  price  at  which  the 
entire  issue  may  be  purchased  to  net  an  effective  rate  other  than 
the  cash  rate,  the  bonds  maturing  at  each  redemption  date  must 
be  considered  separately,  the  several  values  so  obtained  being 
added  to  fmd  the  total  price. 


BOND  DISCOUNT  AND  PREMIUM  215 

As  a  simple  illustration  assume  that  five  bonds  of  $100  each, 
bearing  6%  interest  payable  semiannually,  are  to  be  retired  in 
amounts  of  $100  at  the  end  of  each  of  five  years.  Required — the 
price  to  net  5%.  The  following  values  are  taken  from  a  bond 
table,  though  they  could  be  computed  by  the  methods  already 
explained. 

Maturities  Value 

Bond  due  in  i  year $100.96 

"        "     "    2  years 101.88 

"     "3       "      102.75 

4               103.59 

5  "      104-38 


((        u 


Total  value  of  issue $513.56 

The  following  schedule  shows  the  reduction  of  the  premium 
and  the  serial  redemption  of  the  bonds: 

Cost $513-56 

First  period: 

Coupons  3%  of  $500 $15.00 

Income  2  1/2%  of  $513.56 12.84  2.16 

Second  period: 

Carrying  value $511.40 

Coupons  3%  of  $500 $15.00 

Income  2  1/2%  of  $511. 40 12.78  2.22 


.i« 
First  redemption 100.00 

Third  period: 

Carrying  value $409.18 

Coupons  3%  of  $400 $1 2.00 

Income  2  1/2%  of  $409.18 10.23  1.77 

Fourth  period: 

Carrying  value $407.41 

Coupons  3%  of  $400 $12.00 

Income  2  1/2%  of  $407.41 10.18  1.82 

$405.59 


216         MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 

Second  redemption loo.oo 

Fifth  period: 

Carrying  value $305.59 

Coupons  3%  of  $300 $  Q.oo 

Income  2  1/2%  of  $305.59 7.64  1.36 

Sixth  period: 

Carrying  value $304. 23 

Coupons  3%  of  $300 $  9.00 

Income  2  1/2%  of  $304.23 7.61  1.39 

$302.84 
Third  redemption 100.00 

Seventh  period: 

Carrying  value $202.84 

Coupons  3%  of  $200 $  6.00 

Income  2  1/2%  of  $202.84 5.07  .93 

Eighth  period: 

Carrying  value $201.91 

Coupons  3%  of  $200 $  6.00 

Income  2  1/2%  of  $201.91 5.05  .95 

$200.96 
Fourth  redemption 100.00 

Ninth  period: 

Carrying  value $100.96 

Coupons  3%  of  $100 $  3.00 

Income  2  1/2%  of  $100.96 2.53  ^  .47 

Tenth  period: 

Carrying  value $100.49 

Coupons  3%  of  $100 $  3.00 

Income  2  1/2%  of  $100.49 2.51  .49 

$100.00 
Fifth  redemption 100.00 

o 


CHAPTER  XX 


LEASEHOLDS 


Commuted  Rents 

The  problem  of  determining  the  present  value  of  an  annuity 
arises  when  real  estate  is  leased  and  an  advance  payment  is  made 
covering,  or  applying  on,  a  series  of  rents  which  would  otherwise 
be  paid  at  regular  intervals  in  the  future. 

Let  us  assume  that  it  is  proposed  to  lease  certain  property  on 
January  i,  1918,  for  five  years  at  an  annual  rental  of  $1,000, 
payable  on  January  i  of  each  year.  This  contract  would  require 
the  following  payments : 

Date  Rent 


January  i 
I 
I 
I 
I 


1 91 8 $1,000 

1919 1,000 

1920 1,000 

1 92 1 1,000 

1922 1,000 


The  lessee  desires  to  make  one  payment  on  January  i,  1918, 
covering  the  entire  rental;  and  it  is  agreed  between  the  parties  to 
discount  at  5%  the  payments  which  would  otherwise  be  made  in 
1919,  1920,  192 1  and  1922.  The  single  payment  to  be  made 
(i.e.,  the  present  value  of  the  annuity)  may  be  computed  thus: 


Date 

Due 

Rent 

Symbols 

Present  Value 

January  r,  1918 

I.  1919 

"         I,  1920 

1,  1921 

I,  1922 

Now 

1  year  hence 

2  years 

3  " 

4  " 

li.ooo 
1,000 
1,000 
1,000 
1,000 

J  - 

-  (i   +  <•) 

-  (r   +  <•)' 

-  ( r    +  i)i 

-  (i   +  i)^ 

Si. 000. 000 
952.381 
907.029 
S63.838 
822.702 

Single  payment 

Jj. 545.950 

217 


2l8 


MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 


The  single  payment  can  also  be  computed  thus : 

Present  value  of  first  payment 

Present  value  of  future  payments: 

Present    value    of    an    annuity    of    $i,ooo 
for  four  periods  at  5% 
Present  value  of  $1  due  in  four  years  at 

5% $.822702 

Compound  discount $.177298 

Present     value     annuity     of     $1 

=  $.177298  ^  .05 $3-54596 

$3.54596  X  1,000 

Single  payment 


3,545-96 
54,545-96 


Although  the  single  payment  of  $4,545.96  pays  the  rent  for 
five  years,  the  annual  entries  in  the  accounts  must  show  the 
following : 

1.  Annual  rental  of  $1,000 

2.  Annual  interest  earning  on  the  advance  payment 

3 .  Application  of  advance  payment  to  annual  rent 

4.  Reduction  of  value  of  advance  payment 

The  following  schedule  shows  the  figures  used  in  the  annual 
journal  entries. 


Date 

Debit 
Rent 

Credit 
Interest 

Credit 
Leasehold 

Debit      balance      of 
Leasehold     Account 

li.ooo 
1,000 
1,000 
1,000 
1,000 

I177.30 
136.16 
92.97 
47.62 

$1,000.00 

822.70 
863.84 
907.03 
952.38 

$4. 545. 95 

3,545.95 

2,723.2s 

1,859.41 

952.38 

I,  1920 

Totals 

S.5,000 

1454-05 

$4,545.95 

It  is  manifest  that  the  value  of  a  leasehold  depends  largely 
on  the  rate  of  interest  used.     If  the  above  lease  had  been  com- 


LEASEHOLDS 


219 


muted  on  a  basis  of  4%  its  value  would  have  been  $4,629.90,  or 
$83.95  more  than  it  is  worth  at  5%.  On  a  very  short  lease  the 
difiference  is  not  very  great,  but  on  a  99-year  lease  it  amounts 
to  a  very  large  sum. 

Sublease 

A  similar  problem  arises  when  property  is  subleased.  Let  us 
assume  that  A  leases  certain  property  on  January  i,  191 2,  for  a 
period  of  ten  years  at  an  annual  rental  of  $3,000,  and  that  he 
occupies  the  property  until  December  31,  191 7,  at  which  time 
he  assigns  the  lease  to  B.  Due  to  a  rise  in  land  values,  B  is 
willing  to  assume  the  lease  at  an  annual  rental  of  $5,000.  B 
may,  therefore,  make  the  following  payments: 


Date 

Payment  to  Owner  of  Fee 

Payment  to  A 

$3,000 
3.000 
3.000 
3.000 
3.000 

$2,000 
2,000 

I.  1919 

Or  B  may,  with  A's  consent,  pay  A  a  lump  sum  on  January  i, 
1918,  instead  of  annual  payments.  If  the  future  payments  of  this 
annuity  of  $2,000  are  discounted  at  5%,  the  lump  sum  is  com- 
puted as  follows : 


Present  value  of  first  payment 

Present  value  of  an  annuity  of  $2,000  for  four 

periods  at  5%: 
Present  value  of  an  annuity    of    $1,000    (as 

computed  above)  $3,545.95 
Then  $3,545-95  X  2  = 

Single  payment 


i2,000.00 


7,091.90 


,091.90 


220         MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 

The  following  schedule  shows  the  journal  and  cash  entries 
to  be  made  annually  by  B. 


Date 

Debit 

Credit 

Credit 

Credit  Cash 

Balance  of 

Rent 

Interest 

Leasehold 

(paid  owner) 

Leasehold  Account 

Cost  of  leasehold  .  .  . 

$9,091-90 

January  i,  1918  .... 

$    5.000 

$2,000.00 

$  3,000.00 

7,091.90 

I.  1919 

5.000 

1354-60 

1,645.40 

3,000.00 

5.446.50 

I,  1920  .... 

5.000 

272.33 

1.727.67 

3,000.00 

3.718.83 

I,  1921  .... 

5.000 

185.94 

1,814.06 

3,000.00 

1,904.77 

I,  1922  .... 

5.000 

95-24 

1,904.76 

3,000.00 

.01 

Total 

$25,000 

J908.11 

$9,091-89 

$15,000.00 

CHAPTER  XXI 

DEPRECIATION  METHODS 

Annual  Depreciation 

There  are  six  methods  in  more  or  less  general  use  for  the  de- 
termination of  the  amount  of  depreciation  which  business  en- 
terprises should  provide  annually  on  their  fixed  assets.  Writing 
of^  an  arbitrary  amount  at  any  convenient  time  is  not  included, 
because  it  is  too  haphazard  and  unscientific  to  be  called  a 
method.     The  six  recognized  methods  are  as  follows: 

1.  Straight  line  method 

2.  Diminishing  value,  using  a  fixed  per  cent 

3.  Diminishing  value  method  based  on  the  sum  of  the  years' 

digits 

4.  Annuity  method 

5.  Sinking  fund  method 

6.  Production  method 

It  is  not  within  the  province  of  this  book  to  examine  the  merits 
of  the  different  methods.  The  object  is  only  to  indicate  how  the 
amount  to  be  written  off  annually  is  ascertained  when  the 
managers  of  the  business  have  decided  upon  the  method  to  be 
used. 

The  illustration  to  be  used  throughout  is  that  of  an  asset 
costing  $5 ,000  which  is  estimated  to  have  a  life  of  six  years  and  a 
residual  or  scrap  value  of  $200,  if  any. 

I .  The  straight  line  method  has  the  advantage  of  simplicity, 
as  the  annual  amount  is  determined  by  the  simple  process  of 
dividing  the  total  depreciation  by  the  number  of  years  the  asset 
is  expected  to  remain  effective,  as  shown  below. 


222 


MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 


Year 

Annual  Depreciation 

Carrying  Value 

I 

$    8oo 
800 
8oo 

800 

800 

8oo 

Is, 000 
4,200 
3,400 

2 

4 

s 

6 

1,000 
200 

Total 

S4,8oo 

2.  The  diminishing  value  method,  when  a  fixed  per  cent  is 
applied,  is  a  desirable  one  to  use,  but  it  has  the  disadvantage  that 
the  rate  on  the  diminishing  value  is  difficult  to  compute.  The 
process  requires  the  extraction  of  a  root  the  index  of  which  is  the 
same  as  the  number  of  years  covered  by  the  depreciation. 

The  formula  for  this  method  is 

In  this  formula  r  represents  the  desired  rate;  w  the  number  of 
periods ;  5  the  scrap  value ;  and  c  the  cost.  The  formula  is  applied 
to  the  illustration  as  follows : 

6'  200 

\  5,ooo 

6/ 

=     I  —    V -04 

Log  .04  is 8.60206  —  10 

Add 50.  ~  50 

58.60206  —  60 

Divide  by  6 9.76701  —  10  log  of  .5848 

r=    I-  .5848 
=  .4152 
=  41.52% 

The  following  table  shows  the  depreciation  charges  and  the 
diminishing  value  in  the  example  taken. 


DEPRECIATION  METHODS 


223 


Year 

Depreciation 

Diminishing  value 

I 
2 
3 
4 
5 
6 

41.52%  of  Is, 000. 00 

"     2,924.00 

"     1,709.96 

99998 

584-79 

341-99 

12,076.00 
1,214.04 
709-98 
415-19 
242-80 
141.99 

$5,000.00 

2,924.00 

1,709.96 

999-98 

584-79 

341-99 

200-00 

Total 

$4,800.00 

The  fixed  per  cent  of  diminishing  value  is  very  frequently 
used,  but  because  of  the  difficulty  or  ignorance  of  the  method 
of  computing  it,  the  rate  is  guessed  at,  or  a  purely  arbitrary  one  is 
used,  which,  however,  is  usually  too  small.  Thus,  it  might  seem 
to  many  factory  managers  that  25%  would  be  too  large  a 
rate,  in  the  preceding  illustration  but  it  really  is  not,  as  the  fol- 
lowing shows : 


Year 

25%   OF 
Diminishing  value 

Diminishing  value 

I 

2 

3 

4       , 

S 

6 

$1,250-00 

937-50 
703-13 
527-34 
395-51 
296.63 

$5,000.00 

3.750.00 
2,812.50 
2,109.37 
1,582.03 
1,186.52 
889.89 

Total 

$4,110.1 1 

3.  The  diminishing  value  method  based  on  the  sum  of  the 
years'  digits  is  one  which  is  often  used  to  avoid  the  difficulties  of 
the  percentage  method  described  above.  To  find  the  sum  of  the 
years'  digits  the  figures  representing  the  successive  years  are 
added  together  to  form  a  denominator,  in  our  illustration,  21 


224         MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 

(1  +  2+3+4+5  +  6).  Each  year  a  numerator  is  used  represent- 
ing the  successive  number  of  years  the  asset  is  expected  to  Hve, 
and  the  fraction  thus  obtained  is  apphed  to  the  estimated  total 
depreciation  to  determine  the  amount  to  be  charged  off  that 
year.  The  result  of  applying  this  method  in  our  illustration  is 
as  follows: 


Year 

Fraction 

Annual  Charge 

Carrying  Value 

$5. 000. 00 

I 

6/21 

$1,371.43 

3.628.57 

2 

S/21 

1. 142. 86 

2,485.71 

3 

4/21 

914.29 

1,571.42 

4 

3/21 

685.71 

885.71 

5 

2/21 

457-14 

428.57 

6 

1/21 

228.57 

200.00 

Total .  . . 

21/21 

$4,800.00 

As  is  readily  seen,  this  method  spreads  the  depreciation  with 
less  difference  between  the  early  and  the  late  years  than  does  the 
second  method. 

4.  The  annuity  method,  to  quote  Hatfield 

.  .  .  rests  upon  the  assumption  that  the  cost  of  production  includes  not 
only  repairs  and  the  depreciation  of  machinery,  but  as  well  interest  on  the 
amount  of  capital  invested  in  the  machine.  Depreciation,  on  this  theory, 
should  be  a  sum  figured  as  a  constant  annual  charge,  sufficient  not  only  to 
write  off  the  decline  in  value  but  also  to  write  off  annual  interest  charges 
on  its  diminishing  value. ' 

In  other  words,  this  method  treats  the  cost  of  machinery  as  an 
investment  earning  interest.  Hence  the  cost  of  machinery  is 
dealt  with  as  the  present  value  of  an  annuity;  the  depreciation  to 
be  written  off  periodically  is  an  equal  amount,  and  the  credit  to 
interest  decreases  periodically  because  of  the  diminishing  value 
of  the  asset. 


'  H.  R.  Hatfield,  Modern  Accounting,  191 1.  P-  131. 


DEPRECIATION  METHODS 


225 


If  there  were  no  scrap  value,  the  computation  of  the  annual 
depreciation  charge  would  be  as  follows:  The  cost  of  the  machine 
is  the  present  value  of  an  annuity  of  unknown  rents,  or  depreci- 
ation charges;  this  cost  is  divided  by  the  present  value  of  an 
annuity  with  rents  of  $1.  If  it  is  assumed  that  the  asset  in  our 
illustration  will  have  no  scrap  value  and  that  the  annuity  is  to  be 
based  on  an  interest  rate  of  6%,  the  computation  is  as  follows: 

Present  value  of  $1  due  six  periods  hence.  ...  $          .70406054 

Compound  discount  ($1  —  $.70496054) .29503946 

Present  value  of  annuity  of  $1    ($.29503946 

■^  .06) 4.917324 

Annual  amount  required  ($5,000 -J-  4.917324).  1,016.81 

The  following  is  a  table  of  depreciation  for  our  example, 
based  on  the  annuity  method,  assuming  the  asset  has  no  scrap 
value. 


Year 

Debit 
Depreciation 

Credit 
Interest 

Credit 

Depreciation 

Reserve 

Balance-carrying  value 
Basis  of  interest 

I 
2 
3 
4 
5 
6 

I1.016.81 
i,oi6.8r 
1. 016. 81 
1. 016. 81 
r.oi6.8i 
1,016.81 

$    300.00 
256.99 
21  r.40 
163.08 
111.8s 
57-56 

$    716.81 
759.82 
80s. 41 
853-73 
904.96 
959-25 

Is. 000.00 
4.283.19 
3,52337 
2,717.96 
1,864.23 
959.27 
.02 

Total 

16,100.86 

Si, 100. 88 

14,999-98 

When  the  asset  has  a  scrap  value,  the  conditions  are  more 
complicated,  because  the  cost  of  the  asset  consists  of  two 
elements : 

(a)  The  present  worth  of  an  annuity  of  the  depreciation 

charges,  and 

(b)  The  present  worth  of  the  scrap  value. 


226         MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 

Referring  to  our  illustration,  if  the  asset  has  a  scrap  value  of 
$200,  the  $5,000  invested  in  it  is  the  sum  of  the  following  two 
items : 

(a)  The  present  value  at  6%  of  $200  remaining  after  six 

years 

(b)  The  present  value  on  the  basis  of  6%  of  an  annuity  of 

six  rents  of  unknown  amounts 

The  rents,  or  depreciation  charges,  are  computed  thus: 

Cost  of  asset $5,000.00 

Deduct  present  value  of  $200  due  in  six  years 

(.70496054  X  200) 140.99 

Present  value  of  six  depreciation  charges $4,859.01 

In  the  preceding  illustration  the  present  value  of  an  annuity  of  $1 
for  six  periods  at  6%  was  found  to  be  $4.917324. 

Then  $4,859.01  -r-  4.917324  =  $988.14   annual  depreciation 

The  following  is  the  table  of  depreciation  based  on  the  an- 
nuity method,  when  the  asset  has  a  scrap  value  of  $200. 


Debit 

Credit 

Credit 

Year 

Depreciation 

Interest 

Depreciation  Reserve 

Balance 

Is, 000. 00 

I 

1    988.14 

J    300.00 

$    688.14 

4. 311. 86 

2 

988.14 

258.71 

729-43 

3,582.43 

3 

988.14 

214.9s 

773.19 

2,809.24 

4 

988.14 

168.55 

819.59 

1,989.6s 

S 

988.14 

119.38 

868.76 

1,120.89 

6 

988.14 

67.25 

920.89 

200.00 

Total 

JS.928.84 

Ji, 128.84 

14,800.00 

It  is  not  pertinent  here  to  enter  into  a  discussion  of  the 
propriety  of  including  interest  on  fixed  assets  among  the  manu- 
facturing expenses. 


DEPRECIATION  METHODS  227 

5.  The  sinking  fund  method  is  based  on  the  assumption  that  a 
fund  is  set  aside  to  accumulate  at  compound  interest  with  which 
to  acquire  a  new  asset  when  the  old  one  is  discarded.  It 
assumes  that  the  funds  for  the  purchase  of  the  asset  will  be  pro- 
vided from  two  sources : 

(a)  The  scrap  value  of  the  old  asset 

(b)  The  sinking  fund 

Since  the  fund  accumulates  at  compound  interest,  it  is  a 
sinking  fund  in  the  mathematical  sense,  but  not  in  the  accounting 
sense,  which  limits  the  term  "  sinking  fund  "  to  a  fund  accumu- 
lated to  pay  a  definite  hability.  It  would  be  preferable  to  call 
this  fund  a  replacement  fund  accumulated  on  the  sinking  fund 
principle. 

At  the  end  of  the  life  of  the  asset  the  fund  should  equal  the 
amount  of  the  depreciation.  The  annual  contributions  to  the 
fund  will  be  the  rents  of  an  annuity  which  will  produce  this  total 
depreciation  fund. 

In  our  illustration  the  total  depreciation  is  $4,800,  or  cost 
minus  scrap  value  (c—s).  Hence  the  annual  contribution  to  the 
fund  {SFC)  will  be  $4,800  divided  by  the  amount  of  an  annuity 
of  $1  for  the  given  time  and  at  the  given  rate.  This  is  found  by 
dividing  the  compound  interest  on  $1  for  the  given  time  by  the 
given  rate  of  interest  (I-^i).     The  formula  is: 

SFC  =  {c-  s)^r 
t 

or  SFC  =  (c-  s)X- 

Assuming  that  a  fund  is  to  be  accumulated  on  a  4%  basis,  the 
compound  interest  (/)  on  $1  at  4%  for  six  years  being  $0.265319, 
the  formula  is  applied  thus: 

SFC  =  ($5,000  -  $200)  X      '^"^ 


.265319 


22«         MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 


=    $4,800  X 
=  $192 


.04 


•265319 


.265319 
^723-66 


If  a  fund  is  established,  the  entries  therefor  will  be  a  debit  to 
fund  and  a  credit  to  cash  each  year  for  $723.66.  When  the 
interest  is  collected,  the  cash  goes  into  the  fund  by  an  entry 
debiting  fund  and  crediting  interest.  In  addition  there  must  be 
entries  for  depreciation,  debiting  depreciation  and  crediting 
reserve  for  depreciation  with  an  amount  equal  to  the  sum  of  the 
cash  contributed  and  the  interest  earned  each  year. 

The  following  table  shows  the  operation  of  the  sinking  fund 
method : 


End  of 

Credit 

Credit 

Debit 

Total  Fund 

Carrying 

Year 

Cash 

Interest 

Fund 

and  Reserve 

Value 

$5,000.00 

I 

1    723-66 

$    723-66 

J    723-66 

4.276.34 

3 

723-66 

1  28.9s 

752-61 

1,476.27 

3.523.73 

3 

723-66 

59-05 

782.71 

2,258.98 

2,741.02 

4 

72366 

90.36 

814.02 

3.073-00 

1,927.00 

5 

723.66 

122.92 

846.58 

3.919-58 

1,080.42 

6 

72366 

156.78 

880.44 

4,800.02 

199-98 

Total ... 

$4,341-96 

I458.06 

I4.800.02 

The  annual  charges  to  depreciation  are  the  amounts  in  the 
column  headed  "Debit  Fund."  Thus  the  reserve  for  depreci- 
ation is  always  equal  to  the  fund.  The  charge  to  operations  on 
account  of  depreciation  increases  annually,  but  this  increase  is 
offset  by  the  credit  to  interest,  making  the  net  expense  the  same 
each  year. 

6.  Concerning  the  production  method   ^lontgomery  says: 

A  method  of  making  depreciation  allowances  which  has  its  advantages 
under  certain  conditions  is  that  of  charging  an  established  rate  per  unit  of 


DEPRECIATION  METHODS 


229 


output.  This  is  especially  applicable  in  the  case,  say,  of  a  blast  furnace 
where  the  frequency  with  which  the  linings  will  need  to  be  renewed  de- 
pends on  the  extent  to  which  the  furnace  is  being  used.  If  it  is  being  run 
at  full  capacity  night  and  day,  the  wear  on  the  linings  is  obviously  much 
greater  than  if  the  furnace  had  not  been  in  continual  use  during  the  entire 
fiscal  period.  ^ 

Mr.  Montgomery  applies  this  method  with  perfect  justice  to 
the  lessening  in  value  of  a  wasting  asset,  such  as  timber,  or  the 
coal  or  ore  in  a  mine,  but  this  lessening  in  value  is  not  caused  by 
depreciation,  but  by  an  actual  consumption,  or  removal  and 
conversion  of  the  asset.  In  fact,  this  is  the  only  possible 
method  to  be  applied  to  those  assets  which  diminish  in  exact 
ratio  to  the  amount  used. 

There  can  be  no  rules  formulated  for  the  determination  of  the 
amount  to  be  written  ofif  against  each  unit  of  production.  That 
is  a  matter  that  must  be  left  to  the  judgment  of  the  factory 
managers,  guided  by  experience. 

The  following  is  a  comparative  table  of  depreciation  charges, 
made  according  to  the  first  five  methods : 


Per  Cent  of 

Sum  of 

Straight 

Diminishing 

Years' 

Sinking 

Year 

Line 

Value 

Digits 

Annuity 

Fund 

I 

$    800 

$2,076.00 

$1,371-43 

1    988.14 

$    723.66 

2 

800 

1,214.04 

1,142.86 

988 

14 

752.61 

3 

800 

709-98 

914.29 

988 

14 

782.71 

4 

800 

415-19 

685.71 

988 

14 

814.02 

S 

800 

242.80 

4S7.I4 

988 

14 

846.58 

6 

800 

141.99 

228. S7 

988 

14 

880.44 

Totals 

I4.800 

$4,800.00 

$4,800.00 

$5,928.84 

$4,800.02 

Credit  to  interest 

$1,128.84 

Net 

'  R.  H.  Montgomery,  Auditing  Theory  and  Practice,  1919,  p.  550. 


230 


MATHEMATICS  OF  ACCOUNTING  AND  FINANCE 


The  comparative  carrying  valuts  in  the  same  example  under 
the  five  methods  are  as  follows: 


End  of 
Year 

Straight 
Line 

Per  Cent  of 

Diminishing 

Value 

Sum  of 
Years' 
Digits 

Annuity 

Sinking 
Fund 

$S,ooo 

I5.000.00 

$5,000.00 

$5,000.00 

$5,000.00 

I 

4,200 

2,924.00 

3.628.57 

4.3 1 1  86 

4.276.34 

2 

3.400 

1,709.96 

2,485-71 

3.582.43 

3.523-73 

3 

2,600 

999.98 

1.571-42 

2,809.24 

2.74102 

4 

1,800 

584-79 

885-71 

1,989.65 

■        1,927-00 

S 

1. 000 

341-99 

428.57 

1,120.89 

1,080.42 

6 

200 

200.00 

200.00 

200.00 

199.98 

APPENDIX  A 

VALUES   OF   FOREIGN   COINS 

Following  is  a  list  of  foreign  monetary  units  and  their  values, 
representing  the  pars  of  exchange,  as  estimated  by  the  United 
States  Director  of  the  Mint. 


Value  in 

COUNTRY 

Legal  Standard 

Monetary  Unit 

Terms  of 

U.  S. 

Money 

Argentine  Republic 

Gold 

Peso 

S0.9648 

Gold 

.1930 
.3893 

Bolivia 

Gold 

Boliviano 

Brazil 

Gold 

Milreis 

•  5462 

British  Colonies  in  Austral- 

asia and  Africa 

Gold 

Pound  sterling 

4.866s 

Gold 

Dollar 

Central  American  States: 

Costa  Rica 

Gold 

Colon 

.4653 

British  Honduras 

Gold 

Dollar 

1. 0000 

Gold 

1. 0000 

Guatemala \ 

Honduras / 

Silver 

Peso 

.4403 

Gold 
Gold 

Peso 
Peso 

.5000 

Chile 

■3650 

Amoy 

.7219 

Canton 

■  7197 

Cheefoo 

.6904 

Chin  Kiang 

.7052 

Fuchau 

.6678 

Haikwan 

•  7345 

(customs) 

Hankow 

■6754 

Tael    ■ 

Kiaochow 

Nankin 

Niuchwang 

Ningpo 

Peking 

.6995 
.7143 
.6770 
.6940 
■  7037 

China 

Silver                     J 

Shanghai 
Swatow 
Takau 
^Tientsin 
Yuan 

■  6594 
.6668 
.7264 
.6995 
•4730 

Dollar 

Hongkong 

British 

Mexican 

.4748 
.4748 
.4783 

231 


232 


VALUES  OF  FOREIGN  COINS 


Country 


Legal  Standrd 


Colombia 

Cuba 

Denmark 

Ecuador 

Egypt 

Finland 

France 

Germany 

Great  Britain 

Greece 

Haiti 

India  [British] 

Indo-China 

Italy 

Japan 

Liberia 

Mexico 

Netherlands 

Newfoundland 

Norway 

Panama 

Paraguay 

Persia 

Peru 

Philippine  Islands  . . 

Portugal 

Roumania 

Russia 

Santo  Domingo  .  . .  . 

Serbia  

Siam 

Spain 

Straits  Settlements  . 

Sweden 

Switzerland 

Turkey 

Uruguay  

Venezuela 


Gold 

Gold 
Gold 
Gold 
Gold 

Gold 

Gold  and  silver 

Gold 

Gold 

Gold  and  silver 

Gold 

Gold 

Silver 

Gold 

Gold 

Gold 

Gold 
Gold 
Gold 
Gold 
Gold 
Gold 

(Gold 
\  Silver 

Gold 

Gold 

Gold 

Gold 

Gold 

Gold 

Gold 

Gold 

Gold  and  silver 

Gold 
Gold 
Gold 

Gold 

Gold 
Gold 


Monetary  Unit 


(Dollar)  Peso 

Dollar 

Krone 

Sucre 

Pound  (lOO  piasters) 

Finmark 

Franc 

Mark 

Pound  sterling 

Drachma 

Gourde 

Rupee 

Piaster 

Lira 

Yen 

Dollar 

Peso 

Guilder  (Florin) 

Dollar 

Krone 

Dollar 

Peso  (Argentine) 

Ashrafi 

Kran 

Libra 

Peso 

Escudo 

Leu 

Ruble 

Dollar 

Dinar 

Tical 

Peseta 

Dollar 
Krona 
Franc 

Turkish  Pound 

Peso 
Bolivar 


Value  in 

Terms  of 

U.  S. 

Money 


1. 0000 

.12680 

.4867 

4-94  J I 

•  igjo 
.1930 
.2382 

4.866s 

•  1930 
.2500 
•3244 

•  4755 
.1930 
.4985 

1. 0000 

■  4985 
.4020 

1. 0000 
.2680 

1. 0000 
.9648 

■  0959 
.0811 

4.866s 
.5000 

1.080S 
.1930 
.5146 

r.oooo 
.1930 

•  3709 
.1930 

.5678 
.2680 
.1930 

.0440 

10342 
.1930 


APPENDIX  B 
LOGARITHMS   OF   NUMBERS^ 


No. 

0 

I 

2 

3 

4 

5 

6 

7 

8 

9 

lOO 

00  000 

00  043 

00  087 

00  130 

00  173 

00  217 

00  260 

00  303 

00  346 

00  389 

lOI 

00  432 

00475 

00  518 

00  561 

00  604 

00  647 

00  689 

00  732 

00  775 

00  817 

102 

00  860 

00903 

00  945 

00  988 

01  030 

01  072 

01  IIS 

01  IS7 

01  199 

01  242 

103 

01  284 

01  326 

01  368 

01  410 

01  452 

or  494 

01  536 

01  578 

01  620 

01  662 

104 

01  703 

01  745 

01  787 

01  828 

01  870 

01  912 

01  953 

01  995 

02  036 

02  078 

105 

02  119 

02  160 

02  202 

02  243 

02  284 

02  32s 

02  366 

02  407 

02  449 

02  490 

106 

02  S3  I 

02  572 

02  612 

02  653 

02  694 

02  735 

02  776 

02  816 

02  857 

02  898 

107 

02  938 

02  979 

03  019 

03  060 

03  100 

03  141 

03  181 

03  222 

03  262 

03  302 

108 

03  342 

03  383 

03  423 

03  463 

03  5  03 

03  543 

03  583 

03  623 

03  663 

03  703 

109 

03  743 

03  782 

03  822 

03  862 

03  902 

03  941 

03  981 

04  021 

04  060 

04  100 

IIO 

04  139 

04  179 

04  218 

04  258 

04  297 

04  336 

04  376 

04  4IS 

04  454 

04  493 

III 

04  S32 

04  S7I 

04  610 

04  650 

04  689 

04  727 

04  766 

04  805 

04  844 

04  883 

112 

04  922 

04  961 

04  999 

05  038 

OS  077 

OS  115 

OS  154 

05  192 

05  231 

OS  269 

113 

OS  308 

OS  346 

05  385 

05  423 

OS  461 

OS  500 

OS  538 

OS  576 

OS  614 

OS  652 

114 

05  690 

05  729 

OS  767 

05  80s 

OS  843 

OS  881 

05  918 

OS  956 

OS  994 

06  032 

IIS 

06  070 

06  108 

06  14s 

06  183 

06  221 

06  258 

06  296 

06  333 

06  371 

06  408 

116 

06  446 

06  483 

06  521 

06  SS8 

06  595 

06  633 

06  670 

06  707 

06  744 

06  781 

117 

06  819 

06  856 

06  893 

06  930 

06  967 

07  004 

07  041 

07  078 

07  lis 

07  isi 

118 

07  188 

07  225 

07  262 

07  298 

07  335 

07  372 

07  408 

07  445 

07  482 

07  S18 

119 

07  555 

07  591 

07  628 

07  664 

07  700 

07  737 

07  773 

07  809 

07  846 

07  882 

120 

07  918 

07  954 

07  990 

08  027 

08  063 

08  099 

08  US 

08  171 

08  207 

08  243 

121 

08  279 

08  314 

08  350 

08  386 

08  422 

08  458 

08  493 

08  529 

08  56s 

08  600 

122 

08  636 

08  672 

08  707 

08  743 

08  778 

08  814 

08  849 

08  884 

08  920 

08  9SS 

123 

08  991 

09  026 

09  061 

09  096 

09  132 

09  167 

09  202 

09  237 

09  272 

09  307 

124 

09  342 

09  377 

09  412 

09  447 

09  482 

09  517 

09  552 

09  587 

09  621 

09  656 

125 

09  691 

09  726 

09  760 

09  795 

09  830 

09  864 

09  899 

09  934 

09  968 

10  003 

126 

10  037 

10  072 

10  106 

10  140 

10  175 

10  209 

10  243 

10  278 

10  312 

10  346 

127 

10  380 

10  41S 

10  449 

10  483 

10  S17 

10  551 

10  585 

10  619 

10  653 

10  687 

128 

10  721 

10  755 

10  789 

10  823 

10  857 

10  890 

10  924 

10  958 

10  992 

II  02s 

129 

II  059 

II  093 

II  126 

II  160 

II  193 

II  227 

II  261 

II  294 

II  327 

II  361 

130 

ir  394 

II  428 

II  461 

II  494 

II  528 

II  561 

II  594 

II  628 

II  661 

II  694 

131 

II  727 

1 1  760 

II  793 

II  826 

II  860 

II  893 

1 1  926 

II  959 

II  992 

12  024 

132 

12  057 

12  090 

12  123 

12  156 

12  189 

12  222 

12  254 

12  287 

12  320 

12  352 

133 

12  385 

12  418 

12  450 

12  483 

12  S16 

12  548 

12  S8l 

12  613 

12  646 

12  678 

134 

12  710 

12  743 

12  775 

12  808 

12  840 

12  872 

12  905 

12  937 

12  969 

13  001 

'  E.  H  Barker,  Computing  Tables  and  Formulas.  1913,  pages  22-39. 

233 


234 


APPENDIX 


No. 

0 

I 

2 

3 

4 

5 

6 

7 

8 

9 

135 

13  033 

13  066 

13  098 

13  130 

13  162 

13  194 

13  226 

13  258 

13  290 

13  322 

136 

13  354 

13  386 

13  418 

13  450 

13  481 

13  SI3 

13  S4S 

13  577 

13  609 

13  640 

137 

13  672 

13  704 

13  73S 

13  767 

13  799 

13  830 

13  862 

13  893 

13  92s 

13  956 

138 

13  988 

14  019 

14  051 

14  082 

14  114 

14  14s 

14  176 

14  208 

14  239 

14  270 

139 

14  301 

14  333 

14  364 

14  395 

14  426 

14  457 

14  489 

14  520 

14  551 

14  582 

140 

14  613 

14  644 

14  675 

14  706 

14  737 

14  768 

14  799 

14  829 

14  860 

14  891 

141 

14  922 

14  953 

14  983 

IS  014 

15  045 

IS  076 

IS  106 

IS  137 

IS  168 

IS  198 

142 

15  229 

IS  259 

IS  290 

IS  320 

IS  3SI 

IS  381 

IS  412 

IS  442 

IS  473 

IS  S03 

143 

IS  534 

IS  564 

IS  594 

IS  62s 

15  6s5 

IS  685 

IS  715 

IS  746 

IS  776 

IS  806 

144 

IS  836 

IS  866 

IS  897 

IS  927 

15  957 

IS  987 

16  017 

16  047 

16  077 

16  107 

14s 

16  137 

16  167 

16  197 

16  227 

16  256 

16  286 

16  316 

16  346 

16  376 

16  406 

146 

16  435 

16  46s 

16  495 

16  524 

16  554 

16  584 

16  613 

16  643 

16  673 

16  702 

147 

16  732 

16  761 

16  791 

16  820 

16  850 

16  879 

16  909 

16  938 

16  967 

16  997 

148 

17  026 

17  056 

17  08s 

17  114 

17  143 

17  173 

17  202 

17  231 

17  260 

17  289 

149 

17  319 

17  348 

17  377 

17  406 

17  435 

17  464 

17  493 

17  522 

17  551 

17  580 

ISO 

17  609 

17  638 

17  667 

17  696 

17  72s 

17  754 

17  782 

17  811 

17  840 

17  869 

151 

17  898 

17  926 

17  955 

17  984 

18  013 

18  041 

18  070 

18  099 

18  127 

18  156 

152 

18  184 

18  213 

18  241 

18  270 

18  298 

18  327 

18  355 

18  384 

18  412 

18  441 

153 

18  469 

18  498 

18  526 

18  554 

18  583 

18  611 

18  639 

18  667 

18  696 

18  724 

154 

18  752 

18  780 

18  808 

18  837 

18  865 

18  893 

18  921 

18  949 

18  977 

19  005 

155 

19  033 

19  061 

19  089 

19  1 17 

19  14s 

19  173 

19  201 

19  229 

19  257 

19  285 

156 

19  312 

19  340 

19  368 

19  396 

19  424 

19  451 

19  479 

19  S07 

19  535 

19  562 

157 

19  590 

19  618 

19  645 

19  673 

19  700 

19  728 

19  756 

19  783 

19  811 

19  838 

158 

19  866 

19  893 

19  921 

19  948 

19  976 

20  003 

20  030 

20  058 

20  085 

20  112 

159 

20  140 

20  167 

20  194 

20  222 

20  249 

20  276 

20  303 

20330 

20  3S8 

2038s 

160 

20  412 

20  439 

20  466 

20  493 

20  520 

20  548 

20  575 

20  602 

20  629 

20  656 

161 

20  683 

20  710 

20  737 

20  763 

20  790 

20  817 

20  844 

20  871 

20  898 

2092s 

162 

20952 

20  978 

21  005 

21  032 

21  059 

21  085 

21  112 

21  139 

21  165 

21  192 

163 

21  219 

21  24s 

21  272 

21  299 

21  32s 

21  352 

21  378 

21  405 

21  431 

21  4S8 

164 

21  484 

21  511 

21  537 

21  564 

21  590 

21  617 

21  643 

21  669 

2  1  696 

21  722 

165 

21  748 

21  775 

21  801 

21  827 

21  854 

21  880 

21  906 

21  932 

21  9S8 

21  985 

166 

22  on 

22  037 

22  063 

22  089 

22  IIS 

22  141 

22  167 

22  194 

22  220 

22  246 

167 

22  272 

22  298 

22  324 

22  350 

22  376 

22  401 

22  427 

22  453 

22  479 

22  SOS 

168 

22  531 

22  557 

22  583 

22  608 

22  634 

22  660 

22  686 

22  712 

22  737 

22  763 

169 

22  789 

22  814 

22  840 

22  866 

22  891 

22  917 

22  943 

22  968 

22  994 

23  019 

170 

23  045 

23  070 

23  096 

23  121 

23  147 

23  172 

23  198 

23  223 

23  249 

23  274 

171 

23  300 

23  32s 

23  350 

23  376 

23  401 

23  426 

23  452 

23  477 

23  S02 

23  528 

172 

23  553 

23  578 

23  603 

23  629 

23  654 

23  679 

23  704 

23  729 

23  754 

23  779 

173 

23  80s 

23  830 

23  855 

23  880 

23  90s 

23  930 

23  9S5 

23  980 

24  005 

24030 

174 

24  055 

24  080 

24  105 

24  130 

24  155 

24  180 

24  204 

24  229 

24  254 

24  279 

175 

24  304 

24  329 

24  353 

24  378 

24  403 

24  428 

24  452 

24477 

24  502 

24  527 

176 

24  551 

24  S76 

24  601 

24  62s 

24  650 

24  674 

24699 

24  724 

24  748 

24  773 

177 

24  797 

24  822 

24  846 

24  871 

24  895 

24  920 

24  944 

24  969 

24  993 

25  018 

178 

25  042 

2S  066 

25  091 

25  IIS 

25  139 

25  164 

25  188 

25  212 

25  237 

25  261 

179 

25  28s 

f 

25  310 

25  334 

25  358 

25  382 

25  406 

2S  431 

25  455 

25  479 

25  S03 

LOGARITHMS   OF   NUMBERS 


235 


No. 

0 

I 

2 

3 

4 

5 

6 

7 

8 

9 

180 

25   527 

25  551 

25  575 

25  600 

25624 

25  648 

25  672 

25  696 

25  720 

25  744 

181 

25  768 

25  792 

25  816 

25  840 

25  864 

25  888 

25  912 

25  935 

25  959 

25  983 

182 

26  007 

26  031 

26  05s 

26  079 

26  102 

26  126 

26  150 

26  174 

26  198 

26  221 

183 

26  24s 

26  269 

26  293 

26  316 

26  340 

26  364 

26  387 

26  411 

26  435 

26  458 

184 

26  482 

26  50s 

26  529 

26  553 

26  576 

26  600 

26  623 

26  647 

26  670 

26  694 

I8S 

26  717 

26  741 

26  764 

26  788 

26  811 

26  834 

26  858 

26  881 

26  905 

26  928 

186 

26  951 

26  975 

26  998 

27  021 

27  04s 

27  068 

27  091 

27  114 

27  138 

27  161 

187 

27  184 

27  207 

27  231 

27  254 

27  277 

27  300 

27  323 

27  346 

27  370 

27  393 

188 

27  416 

27  439 

27  462 

27  48s 

27  SO8 

27  531 

27  554 

27  577 

27  600 

27623 

189 

27  646 

27  669 

27  692 

27  715 

27  738 

27  761 

27  784 

27  807 

27  830 

27  852 

190 

27  87s 

27  898 

27  921 

27  944 

27  967 

27  989 

28  012 

28  035 

28  058 

28  081 

191 

28  103 

28  126 

28  149 

28  171 

28  194 

28  217 

28  240 

28  262 

28  285 

28  307 

192 

28  330 

28  353 

2837s 

28  398 

28  421 

28  443 

28  466 

28  488 

28  SIX 

28  533 

193 

28  5S6 

28  578 

28  601 

28  623 

28  646 

28  668 

28  691 

28  713 

28  735 

28  758 

194 

28  780 

28  803 

28  82s 

28  847 

28  870 

28  892 

28  914 

28  937 

28  959 

28  981 

195 

29  003 

29  026 

29  048 

29  070 

29  092 

29  lis 

29  137 

29  159 

29  i8r 

29  203 

196 

29  226 

29  248 

29  270 

29  292 

29  314 

29  336 

29358 

29  380 

29403 

29  425 

197 

29447 

29  469 

29491 

29  513 

29  535 

29  557 

29  579 

29  601 

29  623 

29  645 

198 

29  667 

29  688 

29  710 

29  732 

29  754 

29  776 

29  798 

29  820 

29  842 

29  863 

199 

29  88s 

29  907 

29  929 

29  951 

29  973 

29  994 

30  016 

30  038 

30  060 

30  081 

200 

30  103 

30  125 

30  146 

30  168 

30  190 

30  211 

30  233 

30  255 

30  276 

30  298 

201 

30  320 

30341 

30363 

30384 

30  406 

30  428 

30  449 

30  471 

30492 

30  S14 

202 

30  535 

30  557 

30578 

30  600 

30  621 

30  643 

30  664 

30  685 

30  707 

30  728 

203 

30  750 

30  771 

30  792 

30  814 

30  835 

30  856 

30  878 

30  899 

30  920 

30  942 

204 

30  963 

30  984 

31  006 

31  027 

31  048 

31  069 

31  09X 

31  112 

31  133 

31  154 

20s 

31  175 

31  197 

31  218 

31  239 

31  260 

31  281 

31  302 

31  323 

31  345 

31366 

206 

31  387 

31  408 

31  429 

31  450 

31  471 

31  492 

31  S13 

31  534 

31  555 

31  576 

207 

31  597 

31  618 

31  639 

31  660 

31  681 

31  702 

31  723 

31  744 

31  765 

31  78s 

208 

31  806 

31  827 

31  848 

31  869 

31  890 

31  911 

31  931 

31  952 

31  973 

31  994 

209 

32  015 

32  035 

32  056 

32  077 

32  098 

32  118 

32  139 

32  160 

32  181 

32  201 

210 

32  222 

32  243 

32  263 

32   284 

32  305 

32  325 

32  346 

32  366 

32  387 

32  408 

211 

32  428 

32  449 

32  469 

32  490 

32  510 

32  531 

32  552 

32  572 

32  593 

32  613 

212 

32  634 

32  654 

i2   67s 

32  695 

32  715 

32  736 

32  756 

32   777 

32   797 

32  818 

213 

32  838 

32  858 

32  879 

32  899 

32  919 

32  940 

32  960 

32  980 

33  001 

33  021 

214 

a  041 

a  062 

3i  082 

33   102 

33   122 

33   143 

33   163 

33   183 

33   203 

33   224 

215 

a  244 

a  264 

33  284 

33  304 

33  325 

33  345 

33  36s 

33  385 

33  405 

33  42s 

216 

a  445 

3i  465 

33  486 

33   S06 

33   526 

33  546 

33   566 

33  586 

33  606 

33  626 

217 

3i  646 

a  666 

33  686 

33   706 

33   726 

33   746 

33   766 

33   786 

33   806 

33  826 

218 

a  846 

a  866 

33   88s 

33   905 

33  9-25 

33  945 

33  965 

33  985 

34  005 

34  025 

219 

34  044 

34  064 

34  084 

34  104 

34  124 

34  143 

34  163 

34  1S3 

34  203 

34  223 

220 

34  242 

34  262 

34  282 

34  301 

34  321 

34  341 

34  361 

34  380 

34  400 

34  420 

221 

34439 

34  459 

34  479 

34  498 

34518 

34  537 

34  557 

34  577 

34  596 

34616 

222 

34  63s 

34  655 

34674 

34694 

34  713 

34  733 

34  753 

34  772 

34  792 

34  811 

223 

34  830 

34  850 

34  869 

34  889 

34  908 

34  928 

34  947 

34  967 

34986 

35  OOS 

224 

35  025 

35  044 

35  064 

35  083 

35  102 

35  122 

35  141 

35  i6o 

35  180 

35  199 

236 


APPENDIX 


No. 

0 

I 

2 

3 

4 

5 

6 

7 

8 

9 

225 

3S  218 

35  238 

35  257 

35  276 

35  295 

35  31S 

35  334 

35  353 

35  372 

35  392 

226 

35  411 

35  430 

35  449 

35  468 

35  488 

35  507 

35  526 

35  545 

35  564 

35  583 

227 

35  603 

35  622 

35  641 

35  660 

35  679 

35  698 

35  717 

35  736 

35  755 

35  774 

228 

35  793 

35  813 

35  832 

35  8SI 

35  870 

35  889 

35  908 

35  927 

35  946 

35  96s 

229 

35  984 

36  003 

36  021 

36  040 

36  059 

36  078 

36  097 

36  1x6 

36  135 

36  154 

330 

36  173 

36  192 

36  211 

36  229 

36  248 

36  267 

36  286 

36  305 

36  324 

36  342 

231 

36  361 

36  380 

36  399 

36  418 

36  436 

36  455 

36  474 

36  493 

36  Sii 

36  530 

232 

36  549 

36  568 

36  586 

36  605 

36  624 

36  642 

36  661 

36  680 

36  698 

36  717 

233 

36  736 

36  754 

36  773 

36  791 

36  810 

36  829 

36  847 

36  866 

36  884 

36  903 

234 

36  922 

36  940 

36959 

36  977 

36996 

37  014 

37  033 

37  051 

37  070 

37  088 

235 

37  107 

37  125 

37  144 

37  162 

37  181 

37  199 

37  218 

37  236 

37  254 

37  273 

236 

37  291 

37  310 

37  328 

37  346 

37  365 

37  383 

37  401 

37  420 

37  438 

37  457 

237 

37  475 

37  493 

37  511 

37  530 

37  548 

37  S66 

37  585 

37  603 

37  621 

37  639 

238 

37  658 

37  676 

37  694 

37  712 

37  731 

37  749 

37  767 

37  785 

37  803 

37  822 

239 

37  840 

37  858 

37  876 

37  894 

37  912 

37  931 

37  949 

37  967 

37  98s 

38  003 

240 

38  021 

38  039 

38  057 

38  07S 

38  093 

38  112 

38  130 

38  148 

38  166 

38  184 

241 

38  202 

38  220 

38  238 

38  256 

38  274 

38  292 

38310 

38328 

38346 

38364 

242 

38  382 

38  399 

38  417 

38  435 

38  453 

38  471 

38489 

38  S07 

38  525 

38  543 

243 

38  S6i 

38  578 

38  596 

38614 

38  632 

38  650 

38  668 

38686 

38  703 

38  721 

244 

38  739 

38  757 

38  775 

38  792 

38  810 

38  828 

38  846 

38  863 

38  881 

38  899 

24s 

38  917 

38  934 

38  952 

38  970 

38  987 

39  005 

39  023 

39  041 

39  058 

39  076 

246 

39  094 

39  III 

39  129 

39  146 

39  164 

39  182 

39  199 

39  217 

39  235 

39  252 

247 

39  270 

39  287 

39  305 

39  322 

39  340 

39  358 

39  375 

39  393 

39  410 

39428 

248 

39  445 

39  463 

39  480 

39498 

39  SIS 

39  533 

39  SSO 

39  568 

39  585 

39602 

249 

39  620 

39637 

39  655 

39672 

39  690 

39  707 

39  724 

39  742 

39  759 

39  777 

250 

39  794 

39  811 

39  829 

39  846 

39  863 

39  881 

39  898 

39915 

39  933 

39  950 

2SI 

39  967 

39  985 

40  002 

40  019 

40  037 

40  054 

40  071 

40  088 

40  106 

40  123 

2S2 

40  140 

40  157 

40  175 

40  192 

40  209 

40  226 

40  243 

40  261 

40  278 

40  295 

253 

40  312 

40  329 

40  346 

40  364 

40  381 

40  398 

40  41S 

40  432 

40  449 

40  466 

254 

40  483 

40  500 

40  S18 

40  535 

40  552 

40  569 

40  586 

40  603 

40  620 

40  637 

255 

40  654 

40  671 

40  688 

40  70s 

40  722 

40  739 

40  756 

40  773 

40  790 

40  807 

256 

40  824 

40  841 

40  858 

40  875 

40  892 

40  909 

40  926 

40943 

40  960 

40976 

257 

40  993 

41  010 

41  027 

41  044 

41  061 

41  078 

41  095 

41  II I 

41  128 

41  145 

2S8 

41  162 

41  179 

41  196 

41  212 

41  229 

41  246 

41  263 

41  280 

41  296 

41  313 

259 

41  330 

41  347 

41  363 

41  380 

41  397 

41  414 

41  430 

41  447 

41  464 

41  481 

260 

41  497 

41  514 

41  531 

41  547 

41  564 

41  S8l 

41  597 

41  614 

41  631 

41  647 

261 

41  664 

41  681 

41  697 

41  714 

41  731 

41  747 

41  764 

41  780 

41  797 

41  814 

262 

41  830 

41  847 

41  863 

41  880 

41  896 

41  913 

41  929 

41  946 

41  963 

41  979 

263 

41  996 

42  012 

42  029 

42  045 

42  062 

42  078 

42  095 

42  III 

42  127 

42  144 

264 

42  160 

42  177 

42  193 

42  210 

42  226 

42  243 

42  259 

42  275 

42  292 

42  308 

265 

42  325 

42  341 

42  357 

42  374 

42  390 

42  406 

42  423 

42  439 

42  455 

42  472 

266 

42  488 

42  SO4 

42  521 

42  537 

42  553 

42  570 

42  586 

42  602 

42  619 

42  635 

267 

42  651 

42  667 

42  684 

42  700 

42  716 

42  732 

42  749 

42  765 

42  781 

42  797 

268 

42  813 

42  830 

42  846 

42  862 

42  878 

42  894 

42  911 

42  927 

42  943 

42  959 

269 

42  97S 

42  991 

43  008 

43  024 

43  040 

43  056 

43  072 

43  088 

43  104 

43  120 

LOGARITHMS  OF  NUMBERS 


237 


No. 

0 

I 

2 

3 

4 

5 

6 

7 

8 

9 

270 

43  136 

43  152 

43  169 

43  i8s 

43  201 

43  217 

43  233 

43  249 

43  26s 

43  281 

271 

43  297 

43  313 

43  329 

43  345 

43  361 

43  377 

43  393 

43  409 

43  425 

43  441 

272 

43  457 

43  473 

43  489 

43  505 

43  521 

43  537 

43  553 

43  569 

43  584 

43  600 

273 

43  616 

43  632 

43648 

43  664 

43  680 

43  696 

43  712 

43  727 

43  743 

43  759 

274 

43  775 

43  791 

43  807 

43  823 

43  838 

43  854 

43  870 

43  886 

43  902 

43  917 

27s 

43  933 

43  949 

43  965 

43  981 

43  996 

44  012 

44  028 

44  044 

44  059 

44  075 

276 

44  091 

44  107 

44  122 

44  138 

44  154 

44  170 

44  i8s 

44  201 

44  217 

44  232 

277 

44  248 

44  264 

44  279 

44  295 

44  311 

44  326 

44  342 

44  358 

44  373 

44389 

278 

44  404 

44  420 

44  436 

44  451 

44  467 

44  483 

44  498 

44  514 

44  529 

44  545 

279 

44  560 

44  576 

44  592 

44  607 

44623 

44638 

44654 

44  669 

4468s 

44  700 

380 

44  716 

44  731 

44  747 

44  762 

44  778 

44  793 

44  809 

44  824 

44  840 

44  855 

281 

44871 

44  886 

44  902 

44  917 

44  932 

44  948 

44  963 

44  979 

44  994 

45  010 

282 

45  025 

45  040 

45  056 

45  071 

45  086 

45  102 

45  117 

45  133 

45  148 

45  163 

283 

45  179 

45  194 

45  209 

45  22s 

45  240 

45  255 

45  271 

45  286 

45  301 

45  317 

284 

45  332 

45  347 

45  362 

45  378 

45  393 

45  408 

45  423 

45  439 

45  454 

45  469 

28S 

45  484 

45  500 

45  515 

45  530 

45  545 

45  561 

45  576 

45  591 

45  606 

45  621 

286 

45  637 

45  652 

45  667 

45  682 

45  697 

45  712 

45  728 

45  743 

45  758 

45  773 

287 

45  788 

45  803 

45  818 

45  834 

45  849 

45  864 

45  879 

45  894 

45  909 

45  924 

288 

45  939 

45  954 

45  969 

45  984 

46  000 

46  015 

46  030 

46  045 

46  060 

46  075 

289 

46  090 

46  105 

46  120 

46  135 

46  150 

46  165 

46  180 

46  195 

46  210 

46  22s 

290 

46  240 

46  255 

46  270 

46  28s 

46  300 

46  315 

46  330 

46  345 

46  359 

46374 

291 

46  389 

46  404 

46  419 

46  434 

46  449 

46  464 

46  479 

46  494 

46  S09 

46  523 

292 

46  538 

46  553 

46  568 

46583 

46  598 

46  613 

46  627 

46  642 

46  657 

46  672 

293 

46  687 

46  702 

46  716 

46  731 

46  746 

46  761 

46  776 

46  790 

46  805 

46  820 

294 

46  835 

46  850 

46  864 

46  879 

46  894 

46  909 

46  923 

46  938 

46  953 

46  967 

295 

46  982 

46  997 

47  012 

47  026 

47  041 

47  056 

47  070 

47  085 

47  100 

47  114 

296 

47  129 

47  144 

47  159 

47  173 

47  188 

47  202 

47  217 

47  232 

47  246 

47  261 

297 

47  276 

47  290 

47  305 

47  319 

47  334 

47  349 

47363 

47  378 

47  392 

47  407 

298 

47  422 

47  436 

47  451 

47  465 

47  480 

47  494 

47  509 

47  524 

47  538 

47  553 

299 

47  567 

47  582 

47  596 

47  611 

47  62s 

47  640 

47  6S4 

47  669 

47  683 

47  698 

300 

47  712 

47  727 

47  741 

47  7S6 

47  770 

47  784 

47  799 

47  813 

47  828 

47  842 

301 

47  857 

47  871 

47  88s 

47  900 

47  914 

47  929 

47  943 

47  958 

47  972 

47  986 

302 

48  001 

48  015 

48  029 

48  044 

48  058 

48  073 

48  087 

48  lOI 

48  116 

48  130 

303 

48  144 

48  159 

48  173 

48  187 

48  202 

48  216 

48  230 

48  244 

48  259 

48  273 

304 

48  287 

48  302 

48316 

48  330 

48  344 

48  359 

48  373 

48  387 

48  401 

48  416 

30s 

48  430 

48  444 

48  4S8 

48  473 

48487 

48  SOI 

48  S15 

48  530 

48  544 

48  5S8 

306 

48  572 

48  586 

48  601 

48  61S 

48  629 

48  643 

48  6S7 

48671 

48  686 

48  700 

307 

48  714 

48  728 

48  742 

48  756 

48  770 

48  785 

48  799 

48  813 

48  827 

48  841 

308 

48  85s 

48  869 

48  883 

48  897 

48  911 

48  926 

48  940 

48  954 

48  968 

48  982 

309 

48  996 

49  010 

49  024 

49  038 

49  052 

49  066 

49  080 

49  094 

49  loS 

49  122 

310 

49  136 

49  ISO 

49  164 

49  178 

49  192 

49  206 

49  220 

49  234 

49  248 

49  262 

311 

49  276 

49  290 

49304 

49  318 

49332 

49  346 

49  360 

49  374 

49388 

49  402 

312 

4941S 

49429 

49  443 

49  457 

49  47  r 

4948s 

49  499 

49  513 

49  527 

49  541 

313 

49  554 

49  568 

49  582 

49  596 

49  610 

49  624 

49638 

49  651 

49  665 

49679 

314 

49O93 

49  707 

49  721 

49  734 

49  748 

49  762 

49  776 

49  790 

49  803 

49  817 

238 


APPENDIX 


No. 

0 

, 

2 

3 

4 

5 

6 

7 

8 

9 

315 

49  831 

49  84s 

49  859 

49  872 

49  886 

49  900 

49  914 

49  927 

49  941 

49  9SS 

316 

49  969 

49  982 

49  996 

50  010 

SO  024 

50037 

50  051 

50  065 

50  079 

SO  092 

317 

50  106 

50  120 

SO  133 

SO  147 

50  161 

so  174 

SO  188 

SO  202 

50  215 

50  229 

318 

SO  243 

50  256 

SO  270 

50  284 

SO  297 

50311 

SO  325 

SO  338 

50352 

5036s 

319 

50379 

50  393 

50  406 

50  420 

50  433 

50  447 

SO  461 

50474 

50  488 

50  SOI 

320 

50  515 

50  529 

50  542 

50  556 

50  569 

50  583 

50  596 

50  610 

50623 

SO  637 

321 

50651 

50664 

50678 

50691 

SO  705 

50  718 

50  732 

50  745 

so  759 

50  772 

322 

50  786 

50  799 

SO  813 

so  826 

50  840 

50  853 

SO  866 

SO  880 

SO  893 

50907 

323 

50  920 

50934 

SO  947 

SO  961 

SO  974 

so  987 

SI  001 

51  014 

51  028 

51  041 

324 

51  055 

51  068 

51  081 

51  095 

SI  108 

SI  121 

51  135 

51  148 

51  162 

SI  175 

325 

51  188 

51  202 

SI  215 

SI  228 

SI  242 

51  255 

SI  268 

SI  282 

51  295 

51  308 

326 

51  322 

51  335 

51  348 

51  362 

51  375 

SI  388 

51  402 

SI  41S 

SI  428 

SI  441 

327 

51  455 

51  468 

SI  481 

51  495 

51  S08 

51  521 

51  534 

SI  548 

51  561 

51  574 

328 

51  587 

51  601 

51  614 

51  627 

SI  640 

51  6S4 

SI  667 

51  680 

51  693 

51  706 

329 

SI  720 

51  733 

51  746 

51  759 

51  772 

51  786 

51  799 

51  812 

SI  825 

SI  838 

330 

51  851 

SI  865 

51  878 

SI  891 

SI  904 

51  917 

51  930 

51  943 

51  957 

SI  970 

331 

51  983 

SI  996 

5  2  009 

52  022 

52  035 

52  048 

52  061 

52  075 

52  088 

52  lOI 

332 

52  114 

52  127 

52  140 

52  153 

52  166 

52  179 

52  192 

52  205 

52  218 

52  231 

333 

52  244 

52  257 

52  270 

52  284 

52  297 

52  310 

52  323 

52  336 

52  349 

52  362 

334 

52  375 

52  388 

52  401 

52  414 

52  427 

52  440 

52  453 

52  466 

52  479 

52  492 

335 

5  2  504 

52  517 

52  530 

52  543 

52  556 

52  569 

52  582 

52  595 

52  608 

52  621 

336 

52  634 

52  647 

52  600 

52  673 

52  686 

52  699 

52  711 

52  724 

52  737 

52  750 

337 

52  763 

52  776 

52  789 

52  802 

52  81S 

52  827 

52  840 

52  853 

52  866 

52  879 

338 

52  892 

52  905 

52  917 

52  930 

52  943 

52  956 

52  969 

52  982 

52  994 

53  007 

339 

53  020 

53  033 

53  046 

S3  058 

S3  071 

53  084 

53  097 

53   IIO 

53  122 

53  135 

340 

53  148 

53  161 

53  173 

53  186 

53   199 

53  212 

53  224 

S3  237 

53   250 

53  263 

341 

53  275 

53  288 

53  301 

53  314 

53  326 

53  339 

53  352 

53364 

S3  377 

53  390 

342 

53  403 

53  41S 

53  428 

53  441 

53  453 

53  466 

S3  479 

53  491 

53  504 

53  517 

343 

53  529 

53  542 

53  555 

53  567 

53  580 

53  593 

53  605 

53618 

53  631 

53  643 

344 

53  656 

53  668 

53681 

S3  694 

53  706 

53  719 

53  732 

53  744 

53  757 

53  769 

345 

53  782 

53  794 

53  807 

53  820 

53  832 

53  84s 

53  857 

53  870 

53  882 

53  895 

346 

53  908 

53  920 

53  933 

S3  945 

53  958 

53   970 

53  983 

53   995 

54  008 

54  020 

347 

54  033 

54  045 

54  058 

54  070 

54  083 

54  095 

54  108 

54  120 

54  133 

54  145 

348 

54  158 

54  170 

54  183 

54  195 

54  208 

54  220 

54  233 

54  245 

54  258 

54  270 

349 

54  283 

54  295 

54  307 

54  320 

54  332 

54  345 

54  357 

54  370 

54382 

54  394 

350 

54  407 

54  419 

54  432 

54  444 

54  456 

54  469 

54  481 

54  494 

54  S06 

54  518 

351 

54  531 

54  543 

54  555 

54  568 

54  580 

54  593 

54  60s 

54  617 

54  630 

54  642 

352 

54  654 

54667 

54679 

54  691 

54  704 

54  716 

54  728 

54  741 

54  753 

54  76s 

353 

54  777 

54  790 

54  802 

54  814 

54  827 

54  839 

54  851 

54  864 

54  876 

54  888 

354 

54  900 

54  913 

54925 

54  937 

54  949 

54  962 

54  974 

54  986 

54  998 

55  on 

355 

55  023 

55  035 

55  047 

55  060 

55  072 

55  084 

55  096 

55  108 

55  121 

55  133 

356 

55  145 

55  157 

55  169 

55  182 

55  194 

55  206 

55  218 

55  230 

55  242 

55  255 

357 

55  267 

55  279 

55  291 

55  303 

55  315 

55  328 

55  340 

55  352 

55  364 

55  376 

358 

55  388 

55  400 

55  413 

55  42s 

55  437 

55  449 

55  461 

55  473 

55  485 

55  497 

359 

55  509 

55  522 

55  534 

55  546 

55  SS8 

55  570 

55  582 

55  594 

55  606 

55  618 

LOGARITHMS   OP   NUMBERS 


239 


No. 

0 

I 

2 

3 

4 

5 

6 

7 

8 

9 

360 

55  630 

55  O42 

55  654 

55  666 

55678 

55  691 

55  703 

55  715 

55  727 

55  739 

361 

55  751 

55  763 

55  775 

55  787 

55  799 

55  811 

55  823 

55  83s 

55  847 

55  859 

362 

55  871 

55  883 

55  895 

55  907 

ss  919 

SS  931 

55  943 

55  955 

55  967 

55  979 

363 

55  991 

56  003 

56  015 

56  027 

S6  038 

S6  050 

56  062 

56  074 

56  086 

56  098 

364 

56  no 

56  122 

S6  134 

56  146 

56  158 

56  170 

56  182 

56  194 

S6  205 

56  217 

36s 

S6  229 

S6  241 

56  253 

56  265 

56  277 

S6  289 

56  301 

56  312 

56  324 

S6  336 

366 

56  348 

56  360 

56  372 

56384 

S6  396 

56  407 

S6  419 

56431 

S6  443 

5645s 

367 

S6  467 

56478 

56  490 

56  502 

56  514 

56  526 

56  538 

S6  549 

S6  561 

S6  573 

368 

56  585 

56  597 

56  608 

56  620 

56  632 

S6  644 

56656 

56667 

56  679 

56  691 

369 

S6  703 

56  714 

56  726 

S6  738 

S6  750 

56  761 

56  773 

56  78s 

56  797 

56  808 

370 

56  820 

56  832 

56  844 

S6  855 

56  867 

56  879 

56  891 

S6  902 

56  914 

56  926 

371 

56  937 

S6  949 

56  90 1 

S6  972 

56984 

56  996 

57  008 

57  019 

57  031 

57  043 

372 

57  054 

57  066 

57  078 

57  089 

57  loi 

57  113 

57  124 

57  136 

57  148 

57  159 

373 

57  171 

57  183 

57  194 

57  206 

57  217 

57  229 

57  241 

57  252 

57  264 

57  276 

374 

57  287 

57  299 

57  310 

57  322 

57  334 

57  345 

57  357 

57  368 

57  380 

57  392 

375 

57  403 

57  41S 

57  426 

57  438 

5  7  449 

5  7  461 

57  473 

57  484 

57  496 

57  507 

376 

57  SI9 

57  530 

57  542 

57  553 

57  56s 

57  576 

57  588 

5  7  600 

57  611 

57  623 

377 

57634 

57  646 

57  657 

57  669 

57  680 

57  692 

57  703 

57  715 

57  726 

57  738 

378 

57  749 

57  761 

57  772 

57  784 

S7  795 

57  807 

57  818 

57  830 

57  841 

57  852 

379 

57  864 

57  875 

57  887 

57  898 

57  910 

57  921 

57  933 

57  944 

57  955 

57  967 

380 

57  978 

57  990 

58  001 

58  013 

58  024 

58  035 

S8  047 

58  058 

58  070 

58  081 

381 

58  092 

58  104 

58  IIS 

58  127 

S8  138 

58  149 

58  161 

58  172 

58  184 

58  195 

382 

S8  206 

S8  218 

58  229 

S8  240 

58  252 

58  263 

S8  274 

S8  286 

58  297 

58  309 

383 

58  320 

58331 

58  343 

58  354 

58  36s 

58377 

58  388 

58  399 

58  410 

58422 

384 

58  433 

58  444 

58  456 

58  467 

58  478 

58  490 

58  501 

58  512 

58  524 

58  535 

385 

58  546 

S8  557 

58  569 

S8  580 

S8  591 

S8  602 

S8  614 

58  62s 

58  636 

58  647 

386 

58  659 

S8  670 

S8  681 

S8  692 

58  704 

58  715 

58  726 

58  737 

58  749 

58  760 

387 

58  771 

58  782 

58  794 

58  805 

58  816 

S8  827 

58  838 

58  850 

58  861 

S8  872 

388 

58  883 

S8894 

58  906 

58  917 

58  928 

58  939 

S8  950 

58  961 

58973 

58  984 

389 

58  995 

59  006 

59  017 

59  028 

59  040 

59  051 

59  062 

59  073 

59  084 

59  09s 

390 

59  106 

59  118 

59  129 

59  140 

59  151 

59  162 

59  173 

59  184 

59  I9S 

59  207 

391 

59  218 

59  229 

59  240 

59  251 

59  262 

59  273 

59  284 

59  295 

59  306 

59318 

392 

59  329 

59  340 

59  351 

59  362 

59  373 

59  384 

59  395 

59  406 

59417 

59  428 

393 

59  439 

59  450 

59  461 

59472 

59483 

59  494 

59  506 

59  517 

59  528 

59  539 

394 

59  550 

59  S6i 

59  572 

59  583 

59  S94 

59  605 

59  616 

59  627 

59638 

59  649 

395 

59  660 

59  671 

59  682 

59  693 

59  704 

59  715 

59  726 

59  737 

59  748 

59  759 

396 

59  770 

59  780 

59  791 

59  802 

59  813 

59  824 

59  83s 

59  846 

59  857 

59  868 

397 

59  879 

59  890 

59  901 

59  912 

59  923 

59  934 

59  945 

59  950 

59  966 

59  977 

398 

59  988 

59  999 

60  010 

60  02  1 

60  032 

60  043 

60  054 

60  06s 

60  076 

60  086 

399 

60  097 

60  108 

60  119 

60  130 

60  141 

60  152 

60  163 

60  173 

60  184 

60  19s 

400 

60  206 

60  217 

60  228 

60  239 

60  249 

60  260 

60  271 

60  282 

60  293 

60  304 

401 

60  314 

6032s 

60  336 

60347 

60358 

60  369 

60  379 

60  390 

60  401 

60  412 

402 

60  423 

60  433 

60444 

60  455 

60  466 

60477 

60  487 

60  498 

60  S09 

60  520 

403 

60  531 

60  541 

60  552 

60  563 

60  574 

60  584 

60  595 

60  606 

60  617 

60  627 

404 

60  638 

60  649 

60  660 

60  670 

60  681 

60  692 

60  703 

60  713 

60  724 

60  735 

240 


APPENDIX 


No. 

0 

I 

2 

3 

4 

5 

6 

7 

8 

9 

4CS 

60  746 

60  756 

60  767 

60  778 

60  788 

60  799 

60  810 

60  821 

60  831 

60  842 

406 

60  853 

60  863 

60  874 

60  885 

60  89s 

60  906 

60  917 

60  927 

60938 

60  949 

407 

60  959 

60  970 

60  981 

60  991 

61  002 

61  013 

61  023 

61  034 

61  04s 

61  OSS 

408 

61  066 

61  077 

61  087 

61  098 

61  109 

61  119 

61  130 

61  140 

61  151 

61  162 

409 

61  172 

61  183 

61  194 

61  204 

61  215 

61  225 

61  236 

61  247 

61  257 

61  268 

410 

61  278 

61  289 

61  300 

61  310 

61  321 

61  331 

61  342 

61  352 

61  363 

61  374 

411 

61  384 

61  395 

61  405 

61  416 

61  426 

61  437 

61  448 

61  458 

61  469 

61  479 

412 

61  490 

61  500 

61  511 

61  521 

61  532 

61  542 

61  553 

61  563 

61  574 

61  584 

413 

61  595 

61  606 

61  616 

61  627 

61  637 

61  648 

61  638 

6r  669 

61  679 

61  690 

414 

61  700 

61  711 

61  721 

61  731 

61  742 

61  752 

61  763 

61  773 

61  784 

61  794 

415 

61  80s 

61  815 

61  826 

61  836 

61  847 

61  8S7 

61  868 

61  878 

61  888 

61  899 

416 

61  909 

61  920 

61  930 

61  941 

61  951 

61  962 

61  972 

61  982 

61  993 

62  003 

417 

62  014 

62  024 

62  034 

62  04s 

62  055 

62  066 

62  076 

62  086 

62  097 

62  107 

418 

62  118 

62  128 

62  138 

62  149 

62  159 

62  170 

62  180 

62  190 

62  201 

62  211 

419 

62  221 

62  232 

62  242 

62  252 

62  263 

62  273 

62  284 

62  294 

62  304 

62  315 

420 

62  325 

62  335 

62  346 

62  356 

62  366 

62  377 

62  387 

62  397 

62  408 

62  418 

421 

62  428 

62  439 

62  449 

62  459 

62  469 

62  480 

62  490 

62  SCO 

62  511 

62  521 

422 

62  531 

62  542 

62  552 

62  562 

62  572 

62  583 

62  593 

62  603 

62  613 

62  624 

423 

62  634 

62  644 

62  655 

62  665 

62  675 

62  685 

62  696 

62  706 

62  716 

62  726 

424 

62  737 

62  747 

62  757 

62  767 

62  778 

62  788 

62  798 

62  808 

62  818 

62  829 

42s 

62  839 

62  849 

62  859 

62  870 

62  880 

62  890 

62  900 

62  910 

62  921 

62  931 

426 

62  941 

62  951 

62  961 

62  972 

62  982 

62  992 

63  002 

63  012 

63  022 

63  033 

427 

63  043 

63  053 

63  063 

63  073 

63  083 

63  094 

63  104 

63  114 

63  124 

63  134 

428 

63  144 

63  155 

63  165 

63  175 

63  18S 

63  195 

63  205 

63  215 

63  225 

63  236 

429 

63  246 

63  256 

63  266 

63  276 

63  286 

63  296 

63  306 

63  317 

63  327 

63  337 

430 

63  347 

63  357 

63  367 

63  377 

63  387 

63  397 

63  407 

63  417 

63  428 

63  438 

431 

63  448 

63  4S8 

63  468 

63  478 

63  488 

63  498 

63  508 

63  518 

63  528 

63  538 

432 

63  548 

63  558 

63  568 

63  579 

63  589 

63  599 

63  609 

63  619 

63  629 

63  639 

433 

63  649 

63  659 

63  669 

63  679 

63  689 

63  699 

63  709 

63  719 

63  729 

63  739 

434 

63  749 

63  759 

63  769 

63  779 

63  789 

63  799 

63  809 

63  819 

63  829 

63  839 

435 

63  849 

63  859 

63  869 

63  879 

63  889 

63  899 

63  909 

63  919 

63  929 

63  939 

436 

63  949 

63  959 

63  969 

63  979 

63  988 

63  998 

64  008 

64  018 

64  028 

64  038 

437 

64  048 

64  058 

64  068 

64  078 

64  088 

64  098 

64  108 

64  118 

64  128 

64  137 

438 

64  147 

64  157 

64  167 

64  177 

64  187 

64  197 

64  207 

64  217 

64  227 

64  237 

439 

64  246 

64  256 

64  266 

64  276 

64  286 

64  296 

64  306 

64  316 

64  326 

64  335 

440 

64  345 

64  355 

6436s 

64  375 

64  385 

64  395 

64  404 

64  414 

64  424 

64434 

441 

64  444 

64  454 

64  464 

64473 

64  483 

64  493 

64  503 

64  S13 

64  523 

64  532 

442 

64  542 

64  552 

64  562 

64  572 

64  582 

64  591 

64  601 

64  61 1 

64  621 

64  631 

443 

O4  640 

64  650 

64  660 

64  670 

64  680 

64689 

64  699 

64  709 

64  719 

64  729 

444 

64  738 

64  748 

64  7S8 

64  768 

64  777 

64  787 

64  797 

64  807 

64  816 

64  826 

445 

64  836 

64  846 

64  856 

64  865 

64  875 

64  88s 

64  89s 

64  904 

64  914 

64  924 

446 

64  933 

64  943 

64  953 

64  963 

64  972 

04  982 

04  992 

65  002 

65  on 

6s  021 

447 

6S  031 

65  040 

65  050 

65  060 

65  070 

6S  079 

6S  089 

65  099 

65  108 

6s  n8 

448 

65  128 

6S  137 

65  147 

65  157 

6s  167 

65  176 

65  186 

65  iq6 

65  205 

65  215 

449 

65  225 

65  234 

65  244 

65  254 

6s  263 

6S  273 

65  283 

6s  292 

6s  302 

65  312 

LOGARITHMS  OF   NUMBERS 


241 


No. 

0 

I 

2 

3 

4 

5 

6 

7 

8 

9 

450 

65  321 

65  331 

65  341 

65  350 

65  360 

65  369 

65  379 

65  389 

65  398 

65  408 

45 1 

65  418 

65  427 

65  437 

65  447 

65  456 

6s  466 

65  475 

65  485 

65  495 

65  S04 

452 

65  S14 

65  523 

6s  533 

65  543 

6s  552 

6s  562 

6S  571 

6s  S8i 

65  591 

65  600 

453 

6s  610 

65  619 

65  629 

65  639 

65  648 

65  658 

65  667 

6s  677 

6s  686 

65  696 

454 

6s  706 

65  71S 

65  725 

6S  734 

65  744 

6s  753 

6S  763 

65  772 

65  782 

6s  792 

455 

6s  801 

6s  811 

6s  820 

65  830 

6s  839 

65  849 

6s  858 

6s  868 

65  877 

65  887 

456 

6s  896 

6s  906 

65  916 

65  92s 

65  935 

65  944 

65  954 

6s  963 

6S  973 

6s  982 

457 

65  992 

66  001 

66  on 

66  020 

66  030 

66  039 

66  049 

66  058 

66  068 

66  077 

458 

66  087 

66  096 

66  106 

66  IIS 

66  124 

66  134 

66  143 

66  153 

66  162 

66  172 

459 

66  181 

66  191 

66  200 

66  210 

66  219 

66  229 

66  238 

66  247 

66  257 

66  266 

460 

66  276 

66  28s 

66  295 

66  304 

66  314 

66  323 

66  332 

66  342 

66  351 

66  361 

461 

66  370 

66  380 

66  389 

66  398 

66  408 

66  417 

66  427 

66  436 

66  445 

66  455 

462 

66  464 

66  474 

66483 

66  492 

66  502 

66  511 

66  521 

66  530 

66  539 

66  549 

463 

66  558 

66567 

66  577 

66  586 

66  596 

66  605 

66614 

66  624 

66633 

66642 

464 

66  652 

66661 

66  671 

66  680 

66  689 

66  699 

66  708 

66  717 

66  727 

66  736 

46s 

66  745 

66  755 

66  764 

66  773 

66  783 

66  792 

66  801 

66  811 

66  820 

66  829 

466 

66  839 

66  848 

66  857 

66  867 

66  876 

66  88s 

66  894 

66  904 

66  913 

66  922 

467 

66  932 

66  941 

66  950 

66  960 

66  969 

66  978 

66  987 

66  997 

67  006 

67  ors 

468 

67  025 

67  034 

67  043 

67  052 

67  062 

67  071 

67  080 

67  089 

67  099 

67  108 

469 

67  117 

67  127 

67  136 

67  14s 

67  154 

67  164 

67  173 

67  182 

67  191 

67  201 

470 

67  210 

67  219 

67  228 

67  237 

67  247 

67  256 

67  265 

67  274 

67  284 

67  293 

471 

67  302 

67  311 

67  321 

67  330 

67  339 

67  348 

67  357 

67367 

67  376 

67  385 

472 

67  394 

67  403 

67  413 

67  422 

67  431 

67  440 

67  449 

67  459 

67  468 

67  477 

473 

67  486 

67  495 

67  504 

67  514 

67  523 

67  532 

67  541 

67  550 

67  560 

67  569 

474 

67  578 

67  587 

67  596 

67  60s 

67  614 

67  624 

67633 

67  642 

67  651 

67  660 

475 

67  669 

67  679 

67  688 

67  697 

67  706 

67  71S 

67  724 

67  733 

67  742 

67  752 

476 

67  761 

67  770 

67  779 

67  788 

67  797 

67  806 

67  815 

67  825 

67  834 

67  843 

477 

67852 

67  861 

67  870 

67  879 

67  888 

67  897 

67  906 

67  916 

67  92s 

67  934 

478 

67  943 

67  952 

67  961 

67  970 

67  979 

67  988 

67  997 

68  006 

68  CIS 

68  024 

479 

68  034 

68  043 

68  052 

68  061 

68  070 

68  079 

68  088 

68  097 

68  106 

68  115 

480 

68  124 

68  133 

68  142 

68  isi 

68  160 

68  169 

68  178 

68  187 

68  196 

68  205 

481 

68  215 

68  224 

68  233 

68  242 

68  251 

68  260 

68  269 

68  278 

68  287 

68  296 

482 

68  305 

68  314 

68  323 

68  332 

68  341 

68  350 

68  359 

68  368 

68377 

68  386 

483 

68  395 

68  404 

68  413 

68  422 

68  431 

68  440 

68  449 

68  458 

68  467 

68  476 

484 

68  485 

68  494 

68  S02 

68  sii 

68  520 

68  520 

68  538 

68  547 

68  556 

68  s6s 

48s 

68  574 

68  583 

68  592 

68  601 

68  610 

68  619 

68  628 

68  637 

68  646 

68  655 

486 

68  664 

68673 

68  681 

68  690 

68  699 

68  708 

68  717 

68  726 

68  73S 

68  744 

487 

68  753 

68  762 

68  771 

68  780 

68  789 

68  797 

68  806 

68  815 

68  824 

68  833 

488 

68  842 

68  851 

68  860 

68  869 

68  878 

68  886 

68  895 

68  904 

68  913 

68  922 

489 

68  931 

68  940 

68  949 

68  958 

68  966 

68  975 

68  984 

68  993 

69  002 

69  on 

490 

69  020 

69  028 

60  037 

69  046 

69  055 

69  064 

69  073 

69  082 

69  090 

69  099 

491 

69  108 

69  117 

69  126 

69  135 

69  144 

69  152 

69  161 

69  170 

69  179 

69  188 

492 

69  197 

69  205 

69  214 

69  223 

69  232 

69  241 

69  249 

69  258 

69  267 

69  276 

493 

69  285 

69  294 

69  302 

69  311 

69  320 

69  329 

69338 

69  346 

69  355 

69364 

494 

69  373 

69  381 

69  390 

69399 

69  408  69  417  1 

69  42s 

69  434 

69  443 

69  452 

242 


APPENDIX 


No. 

0 

I 

2 

3 

4 

5 

6 

7 

8 

9 

495 

69  461 

69  469 

69  478 

69  487 

69  496 

69  504 

69  S13 

69  522 

69  531 

69  539 

496 

69  548 

69  557 

69  566 

69  574 

69  583 

69  592 

69  601 

69  609 

69  618 

69  627 

497 

69  636 

69  644 

69  653 

69  662 

69  671 

69  679 

69  688 

69  697 

69  705 

69  714 

498 

69  723 

69  732 

69  740 

69  749 

69  758 

69  767 

69  775 

69  784 

69  793 

69  801 

499 

69  810 

69  819 

69  827 

69  836 

69  84s 

69  854 

69  862 

69  871 

69  880 

69  888 

500 

69  897 

69  906 

69  914 

69923 

69  932 

69  940 

69  949 

69  958 

69  966 

69975 

SOI 

69  984 

69  992 

70  001 

70  010 

70  018 

70  027 

70  036 

70044 

70  053 

70  062 

502 

70  070 

70  079 

70  088 

70  096 

70  105 

70  114 

70  122 

70  131 

70  140 

70  148 

S03 

70  157 

70  165 

70  174 

70  183 

70  191 

70  200 

70  209 

70  217 

70  226 

70  234 

5  04 

70  243 

70  252 

70  260 

70  269 

70  278 

70  286 

70  295 

70303 

70312 

70  321 

S05 

70  329 

70338 

70  346 

70  355 

70  364 

70372 

70  381 

70  389 

70  398 

70  406 

S06 

70415 

70424 

70  432 

70441 

70  449 

70  458 

70  467 

70  475 

70  484 

70492 

S07 

70  501 

70  509 

70  518 

70  526 

70  535 

70  544 

70  552 

70  561 

70  569 

70578 

508 

70  s86 

70  595 

70  603 

70  612 

70  621 

70  629 

70638 

70  646 

70  655 

70  663 

509 

70  672 

70  680 

70  689 

70  697 

70  706 

70  714 

70  723 

70  731 

70  740 

70  749 

510 

70  757 

70  766 

70  774 

70  783 

70  791 

70  800 

70  808 

70  817 

70  82s 

70  834 

511 

70  842 

70  851 

70  859 

70  868 

70  876 

70  885 

70  893 

70  902 

70  910 

70  919 

S12 

70  927 

70  935 

70  944 

70  952 

70  961 

70  969 

70  978 

70  986 

70  995 

71  003 

513 

71  012 

71  020 

71  029 

71  037 

71  046 

71  054 

71  063 

71  071 

71  079 

71  088 

514 

71  096 

"I  105 

71  113 

71  122 

71  130 

71  139 

71  147 

71  155 

71  164 

71  172 

515 

71  181 

71  189 

71  198 

71  206 

71  214 

71  223 

71  231 

71  240 

71  248 

71  257 

S16 

71  26s 

71  273 

71  282 

71  290 

71  299 

71  307 

71  31S 

71  324 

71  332 

71  341 

S17 

71  349 

71  357 

71  366 

71  374 

71 383 

71  391 

71  399 

71  408 

71  416 

71  42s 

518 

71  433 

71  441 

71  450 

71  458 

71  466 

71  475 

71  483 

71  492 

71  500 

71  508 

519 

71  517 

71  525 

71  533 

71  542 

71  550 

71  559 

71  567 

71  575 

71  584 

71  592 

520 

71  600 

71  609 

71  617 

71  625 

71  634 

71  642 

71  650 

71  659 

71  667 

71  675 

S2I 

71  684 

71  692 

71  700 

71  709 

71  717 

71  725 

71  734 

7t  742 

71  750 

71  759 

522 

71  767 

71  775 

71  784 

71  792 

71  800 

71  809 

71  817 

71  82s 

71  834 

71  842 

523 

71  850 

71  858 

71  867 

71  87s 

71 883 

71  892 

71  900 

7  r  908 

71  917 

7  t  92s 

524 

71  933 

71  941 

71  950 

71  958 

71  966 

71  975 

71  983 

71  991 

71  999 

72  008 

525 

72  016 

72  024 

72  032 

72  041 

72  049 

72  057 

72  066 

72  074 

72  082 

72  090 

526 

72  099 

72  107 

72  IIS 

72  123 

72  132 

72  140 

72  148 

72  156 

72  165 

72  173 

527 

72  181 

72  189 

72  198 

72  206 

72  214 

72  222 

72  230 

72  239 

72  247 

72  255 

528 

72  263 

72  272 

72  280 

72  288 

72  296 

72  304 

72  313 

72  321 

72  329 

72  337 

529 

72  346 

72  354 

72  362 

72  370 

72  378 

72  387 

72  395 

72  403 

72  411 

72  419 

530 

72  428 

72  436 

72  444 

72  452 

72  460 

72  469 

72  477 

72  48s 

72  493 

72  SOI 

531 

72  509 

72  518 

72  526 

72  534 

72  542 

72  550 

72  558 

72  567 

72  575 

72  583 

532 

72  591 

72  599 

72  607 

72  616 

72  624 

72  632 

72  640 

72  648 

72  656 

72  66s 

533 

72  673 

72  681 

72  689 

72  697 

72  70s 

72  713 

72  722 

72  730 

72  738 

72  746 

534 

72  754 

72  762 

72  770 

72  779 

72  787 

72  795 

72  803 

72  811 

72  819 

72  827 

535 

72  835 

72  843 

72  852 

72  860 

72  868 

72  876 

72  884 

72  892 

72  900 

72  908 

536 

72  916 

72  925 

72  933 

72  941 

72  949 

72  957 

72  96s 

72  973 

72  981 

72  989 

537 

72  997 

73  006 

73  014 

73  022 

73  030 

73  038 

73  046 

73  054 

73  062 

73  070 

538 

73  078 

73  086 

73  094 

73  102 

73  III 

73  119 

73  127 

73  135 

73  143 

73  151 

539 

73  159 

73  167 

73  175 

73  183 

73  191 

73  199 

73  207 

73  215 

73  223 

73  231 

LOGARITHMS  OF   NUMBERS 


243 


No. 

0 

I 

2 

3 

4 

5 

6 

7 

8 

9 

540 

73  239 

73  247 

73  255 

73  263 

73  272 

73  280 

73  288 

73  296 

73  304 

73  312 

541 

73  320 

73  328 

73  336 

73  344 

73  352 

73  360 

73  368 

73  376 

73  384 

73  392 

542 

73  400 

73  408 

73  416 

73  424 

73  432 

73  440 

73  448 

73  456 

73  464 

73  472 

543 

73  480 

73  488 

73  496 

73  504 

73  512 

73  520 

73  528 

73  536 

73  544 

73  552 

544 

73  560 

73  568 

73  576 

73  584 

73  592 

73  600 

73  608 

73  616 

73  624 

73  632 

545 

73  640 

73  648 

73656 

73  664 

73  672 

73  679 

73  687 

73  695 

73  703 

73  711 

546 

73  719 

73  727 

73  735 

73  743 

73  751 

73  759 

73  767 

73  775 

73  783 

73  791 

547 

73  799 

73  807 

73  815 

73  823 

73  830 

73  838 

73  846 

73  854 

73  862 

73  870 

548 

73  878 

73  886 

73  894 

73  902 

73  910 

73  918 

73  926 

73  933 

73  941 

73  949 

549 

73  957 

73  96s 

73  973 

73  981 

73  989 

73  997 

74  005 

74  013 

74  020 

74  028 

550 

74  036 

74  044 

74  052 

74  060 

74  068 

74  076 

74  084 

74  092 

74  099 

74  107 

551 

74  IIS 

74  123 

74  131 

74  139 

74  147 

74  155 

74  162 

74  170 

74  178 

74  186 

552 

74  194 

74  202 

74  210 

74  218 

74  22s 

74  233 

74  241 

74  249 

74  257 

74  265 

553 

74  273 

74  280 

74  288 

74  296 

74  304 

74  312 

74  320 

74  327 

74  335 

74  343 

554 

74  351 

74  359 

74  367 

74  374 

74  382 

74  390 

74  398 

74  406 

74  414 

74  421 

555 

74  429 

74  437 

74  445 

74  453 

74  461 

74  468 

74  476 

74  484 

74  492 

74  500 

556 

74  507 

74  51S 

74  523 

74  531 

74  539 

74  547 

74  554 

74  562 

74  570 

74  578 

557 

74  586 

74  593 

74  601 

74  609 

74  617 

74  624 

74  632 

74  640 

74648 

74  656 

558 

74663 

74  671 

74  679 

74687 

74  69s 

74  702 

74  710 

74  718 

74  726 

74  733 

559 

74  741 

74  749 

74  757 

74  764 

74  772 

74  780 

74  788 

74  796 

74  803 

74  811 

560 

74  819 

74  827 

74  834 

74  842 

74  850 

74858 

74  865 

74  873 

74  881 

74  889 

561 

74  896 

74  904 

74  912 

74  920 

74  927 

74  935 

74  943 

74  950 

74  958 

74  966 

562 

74  974 

74  981 

74  989 

74  997 

75  005 

75  012 

75  020 

75  028 

75  035 

75  043 

563 

75  051 

75  059 

75  066 

75  074 

75  082 

75  089 

75  097 

75  105 

75  113 

75  120 

564 

75  128 

75  136 

75  143 

75  iSi 

75  159 

75  166 

75  174 

75  182 

75  189 

75  197 

S6S 

75  20s 

75  213 

75  220 

75  228 

75  236 

75  243 

75  251 

75  259 

75  266 

75  274 

566 

75  282 

75  289 

75  297 

75  305 

75  312 

75  320 

75  328 

75  335 

75  343 

75  351 

S67 

75  3S8 

75  366 

75  374 

75  381 

75  389 

75  397 

75  404 

75  412 

75  420 

75  427 

568 

75  435 

75  442 

75  4SO 

75  458 

75  465 

75  473 

75  481 

75  488 

75  496 

75  504 

569 

75  511 

75  519 

75  526 

7S  534 

75  542 

75  549 

75  557 

75  565 

75  572 

75  S8o 

570 

75  587 

75  595 

75  603 

75  610 

75  618 

75  626 

75  633 

75  641 

75  648 

75  656 

571 

75  664 

75  671 

75  679 

75  686 

75  694 

75  702 

75  709 

75  717 

75  724 

75  732 

572 

75  740 

75  747 

75  755 

75  762 

75  770 

75  778 

75  785 

75  793 

75  800 

75  808 

573 

75  815 

75  823 

75  831 

75  838 

75  846 

75  853 

75  861 

75  868 

75  876 

75884 

574 

75  891 

75  899 

75  906 

75  914 

75  921 

75  929 

75  937 

75  944 

75  952 

75  959 

575 

75  967 

75  974 

75  982 

75  989 

75  997 

76  005 

76  012 

76  020 

76  027 

76  035 

576 

76  042 

76  050 

76  OS  7 

76  065 

76  072 

76  080 

76  087 

76  095 

76  103 

76  no 

577 

76  118 

76  125 

76  133 

76  140 

76  148 

76  155 

76  163 

76  170 

76  178 

76  i8s 

578 

76  193 

76  200 

76  208 

76  215 

76  223 

76  230 

76  238 

76  24s 

76  253 

76  260 

5  79 

76  268 

76  275 

76  283 

76  290 

76  298 

76  305 

76  313 

76  320 

76  328 

76  335 

580 

76  343 

76  350 

76  358 

76  365 

76373 

76  380 

76  388 

76  395 

76  403 

76  410 

581 

76  418 

76  42s 

76  433 

76  440 

76448 

76  455 

76  462 

76  470 

76  477 

76485 

582 

76  492 

76  500 

76  507 

76  515 

76  522 

76  530 

76  537 

76  545 

76  552 

76  559 

583 

76  567 

76  574 

76  S82 

76  589 

76  597 

76  604 

76  612 

76  619 

76  626 

76  634 

584 

76  641 

76  649 

76  656 

76  664 

76671 

76  678 

76  686 

76  693 

76  701 

76  708 

244 


APPENDIX 


No. 

0 

I 

2 

3 

4 

5 

6 

7 

8 

9 

S8S 

76  716 

76  723 

76  730 

76  738 

76  745 

76  753 

76  760 

76  768 

76  775 

76  782 

S86 

76  790 

76  797 

76  805 

76  812 

76  819 

76  827 

76834 

76  842 

76  849 

76  856 

S87 

76  864 

76  871 

76  879 

76  886 

76  893 

76  901 

76  908 

76  916 

76  923 

76  930 

S88 

76  938 

76  945 

76  953 

76  960 

76  967 

76  975 

76  982 

76  989 

76  997 

77  004 

S89 

77  012 

77  019 

77  026 

77  034 

77  041 

77  048 

77  056 

77  063 

77  070 

77  078 

590 

77  085 

77  093 

77  100 

77  107 

77  IIS 

77  122 

77  129 

77  137 

77  144 

77  151 

591 

77  159 

77  166 

77  173 

77  I8i 

77  188 

77  195 

77  203 

77  210 

77  217 

77  22s 

592 

77  232 

77  240 

77  247 

77  254 

77  262 

77  269 

77  276 

77  283 

77  291 

77  298 

593 

77  30s 

77  313 

77  320 

77  327 

77  335 

77  342 

77  349 

77357 

77  364 

77  371 

594 

77  379 

77  386 

77  393 

77  401 

77  408 

77  415 

77  422 

77  430 

77  437 

77  444 

595 

77  452 

77  459 

77  466 

77  474 

77  481 

77  488 

77  495 

77  503 

77  510 

77  517 

596 

77  525 

77  532 

77  539 

77  546 

77  554 

77  S6i 

77  568 

77  576 

77  583 

77  590 

597 

77  597 

77  60s 

77  612 

77  619 

77  627 

77  634 

77  641 

77  648 

77  656 

77  663 

598 

77  670 

77  677 

77  68s 

77  692 

77  699 

77  706 

77  714 

77  721 

77  728 

77  735 

599 

77  743 

77  750 

77  757 

77  764 

77  772 

77  779 

77  786 

77  793 

77  801 

77  808 

600 

77  815 

77  822 

77  830 

77  837 

77  844 

77  851 

77  859 

77  866 

77  873 

77  880 

6or 

77  887 

77  895 

77  902 

77  909 

77  916 

77  924 

77  931 

77  938 

77  945 

77  952 

602 

77  960 

77  967 

77  974 

77  981 

77  988 

77  996 

78  003 

78  oro 

78  017 

78  025 

603 

78  032 

78  039 

78  046 

78  053 

78  061 

78  068 

78  075 

78  082 

78  089 

78  097 

604 

78  104 

78  III 

78  118 

78  125 

78  132 

78  140 

78  147 

78  IS4 

78  161 

78  168 

605 

78  176 

78  183 

78  190 

78  197 

78  204 

78  211 

78  219 

78  226 

78  233 

78  240 

606 

78  247 

78  254 

78  262 

78  269 

78  276 

78  283 

78  290 

78  297 

78  30s 

78  312 

607 

78  319 

78  326 

78  333 

78  340 

78  347 

78355 

78362 

78  369 

78  376 

78  383 

608 

78  390 

78  398 

78  40s 

78  412 

78  419 

78  426 

78  433 

78  440 

78  447 

78  455 

609 

78  462 

78469 

78  476 

78  483 

78  490 

78  497 

78  S04 

78  512 

78  S19 

78  526 

610 

78  533 

78  540 

78  547 

78  554 

78  S6i 

78  569 

78  576 

78  583 

78  590 

78  597 

611 

78  604 

78  6ri 

78  618 

78  62s 

78  633 

78  640 

78  647 

78  6S4 

78  661 

78668 

612 

78  675 

78  682 

78  689 

78  696 

78  704 

78  711 

78  718 

78  72s 

78  732 

78  739 

613 

78  746 

78  753 

78  760 

78  767 

78  774 

78  781 

78  789 

78  796 

78  803 

78  810 

614 

78  817 

78  824 

78  831 

78  838 

78  84s 

78  852 

78  8S9 

78  866 

78  873 

78  880 

615 

78  888 

78  89s 

78  902 

78  909 

78  916 

78  923 

78  930 

78  937 

78  944 

78  951 

616 

78  958 

78  965 

78  972 

78  979 

78  986 

78  993 

79  000 

79  007 

79  014 

79  021 

617 

79  029 

79  036 

79  043 

79  O.SO 

79  OS 7 

79  064 

79  071 

79  078 

79  08s 

79  092 

618 

79  099 

79  106 

79  113 

79  120 

79  127 

79  134 

79  141 

79  148 

79  155 

79  162 

619 

79  169 

79  176 

79  183 

79  190 

79  197 

79  204 

79  211 

79  218 

79  225 

79  232 

620 

79  239 

79  246 

79  253 

79  260 

79  267 

79  274 

79  281 

79  288 

79  295 

79  302 

621 

79  309 

79  316 

79  323 

79  330 

79  337 

79  344 

79  351 

79  358 

79  36s 

79372 

622 

79  379 

79386 

79  393 

79  400 

79  407 

79  414 

79  421 

79  428 

79  435 

79  442 

623 

79  449 

79  456 

79  463 

79  470 

79  477 

79  484 

79  491 

79  498 

79  50s 

79  511 

624 

79  518 

79525 

79  532 

79  539 

79  546 

79  553 

79  S6o 

79  567 

79  574 

79  581 

62s 

79  588 

79  595 

79  602 

79  609 

79  616 

79  623 

79  630 

79  637 

79  644 

79  650 

626 

79657 

79  664 

79  671 

79678 

7968s 

79  692 

79  699 

79  706 

79  713 

79  720 

627 

79  727 

79  734 

79  741 

79  748 

79  754 

79  761 

79  768 

79  775 

79  782 

79  789 

628 

79  796 

79  803 

79  810 

79  817 

79  824 

79  831 

79  837 

79  844 

79  851 

79  858 

629 

79  86s 

79  872 

79  879 

79  886 

79  893 

79  900 

79  906 

79  913 

79  920 

79  927 

LOGARITHMS  OF  NUMBERS 


245 


No. 

0 

I 

2 

3 

4 

5 

6 

7 

8 

9 

630 

79  934 

79  941 

79  948 

79  955 

79  962 

79  969 

79  975 

79  982 

79  989 

79  996 

631 

80  003 

80  010 

80  017 

80  024 

80  030 

80  037 

80  044 

80  051 

80  058 

80  06s 

632 

80  072 

80  079 

80  08s 

80  092 

80  099 

80  106 

80  113 

80  120 

80  127 

80  134 

633 

80  140 

80  147 

80  154 

80  161 

80  168 

80  175 

80  182 

80  188 

80  195 

80  202 

634 

80  209 

80  216 

80  223 

80  229 

80  236 

80  243 

80  250 

80  257 

80  264 

80  271 

63  s 

80  277 

80  284 

80  291 

80  298 

80  30s 

80  312 

80  318 

80325 

80332 

80339 

636 

80  346 

80353 

80359 

80  366 

80373 

80  380 

80  387 

80  393 

80  400 

80  407 

637 

80  414 

80  421 

80  428 

80  434 

80  441 

80  448 

80  455 

80  462 

80  468 

80  475 

638 

80  482 

80  489 

80  496 

80  502 

80  509 

80  516 

80  523 

80  530 

80  536 

80  543 

639 

80  550 

80  557 

80  564 

80  570 

80577 

80  584 

80  591 

80  598 

80  604 

80  611 

640 

80  618 

80  625 

80  632 

80  638 

80  64s 

80  652 

80  659 

80  66s 

80  672 

80  679 

641 

80  686 

80  693 

80  699 

80  706 

80  713 

80  720 

80  726 

80  733 

80  740 

80  747 

642 

80  754 

80  760 

80  767 

80  774 

80  781 

80  787 

80  794 

80  801 

80  808 

80  814 

643 

80  821 

80  828 

80  835 

80  841 

80  848 

80  85s 

80  862 

80  868 

80  875 

80  882 

644 

80  889 

80  895 

80  902 

80  909 

80  916 

80  922 

80  929 

80  936 

80943 

80  949 

64s 

80  956 

80  963 

80  969 

80  976 

80  983 

80  990 

80  996 

81  003 

81  010 

81  017 

646 

81  023 

81  030 

81  037 

81  043 

81  050 

81  057 

81  064 

81  070 

81  077 

81  084 

647 

8r  090 

81  097 

81  104 

81  III 

81  117 

81  124 

81  131 

81  137 

81  144 

81  151 

648 

81  158 

81  164 

81  171 

81  178 

81  184 

81  191 

81  198 

81  204 

81  211 

81  218 

649 

81  224 

81  231 

81  238 

81  245 

81  251 

81  258 

81  265 

81  271 

81  278 

81  28s 

650 

81  291 

81  298 

81  305 

81  311 

81  318 

81  325 

81  331 

81  338 

81  345 

81  351 

6s  I 

81  3S8 

81  365 

81  371 

81  378 

81  385 

81  391 

81  398 

81  405 

81  411 

81  418 

652 

81  42s 

81  431 

81  438 

81  445 

81  451 

81  458 

81  465 

81  471 

81  478 

81  485 

653 

81  491 

81  498 

81  505 

81  511 

81  518 

81  525 

81  531 

81  538 

81  544 

81  551 

654 

8r  558 

81  564 

81  571 

81  578 

81  584 

81  591 

81  598 

81  604 

81  611 

81  617 

655 

81  624 

81  631 

81  637 

81  644 

81  651 

81  657 

81  664 

81  671 

81  677 

81  684 

656 

81  690 

8i  697 

81  704 

81  710 

81  717 

81  723 

81  730 

81  737 

81  743 

81  7SO 

6S7 

81  757 

81  763 

81  770 

81  776 

81  783 

81  790 

81  796 

81  803 

81  809 

81  816 

658 

81  823 

81  829 

81  836 

81  842 

81  849 

81  856 

81  862 

8i  869 

81  875 

81  882 

659 

81  889 

81  895 

81  902 

81  908 

81  91S 

81  921 

81  928 

81  935 

81  941 

81  948 

660 

8r  954 

81  961 

81  968 

81  974 

81  981 

81  987 

81  994 

82  000 

82  007 

82  014 

661 

82  020 

82  027 

82  033 

82  040 

82  046 

82  053 

82  060 

82  066 

82  073 

82  079 

662 

82  086 

82  092 

82  099 

82  105 

82  112 

82  119 

82  125 

82  132 

82  138 

82  145 

663 

82  151 

82  158 

82  164 

82  171 

82  178 

82  184 

82  191 

82  197 

82  204 

82  210 

664 

82  217 

82  223 

82  230 

82  236 

82  243 

82  249 

82  256 

82  263 

82  269 

82  276 

66s 

82  282 

82  289 

82  295 

82  302 

82  308 

82  31S 

82  321 

82  328 

82  334 

82  341 

666 

82  347 

82  354 

83  360 

82  367 

82  373 

82  380 

82  387 

82  393 

82  400 

82  406 

667 

82  413 

82  419 

82  426 

82  432 

82  439 

82  445 

82  452 

82  458 

82  465 

82  471 

668 

82  478 

82  484 

82  491 

82  497 

82  504 

82  510 

82  517 

82  52J 

82  530 

82  536 

669 

82  543 

82  549 

82  556 

82  562 

82  569 

82  575 

82  582 

82  s88 

82  595 

82  6or 

670 

82  607 

82  614 

82  620 

82  627 

82  633 

82  640 

82  646 

82  6s3 

82  659 

82  666 

671 

82  672 

82  679 

82  685 

82  692 

82  698 

82  70s 

82  711 

82  718 

82  724 

82  730 

672 

82  737 

82  743 

82  750 

82  7S6 

82  763 

82  769 

82  776 

82  782 

82  789 

82  795 

673 

82  802 

82  808 

82  814 

82  821 

82  827 

82  834 

82  840 

82  847 

82  8S3 

82  860 

674 

82  866 

82  872 

82  879 

82  88s 

82  892 

82  898 

82  90s 

82  911 

82  918 

82  924 

246 


APPENDIX 


No. 

0 

I 

2 

3 

4 

5 

6 

7 

8 

9 

675 

82  930 

82  937 

82  943 

82  950 

82  956 

82  963 

82  969 

82  975 

82  982 

82  988 

676 

82  995 

83  001 

83  008 

83  014 

83  020 

83  027 

83  033 

83  040 

83  046 

83  052 

677 

83  0S9 

83  06s 

83  072 

83  078 

83  085 

83  091 

83  097 

83  104 

83  no 

83  117 

678 

83  123 

83  129 

83  136 

83  142 

83  149 

83  155 

83  161 

83  168 

83  174 

83  181 

679 

83  187 

83  193 

83  200 

83  206 

83  213 

83  219 

83  22s 

83  232 

83  238 

83  24s 

680 

83  251 

83  257 

83  264 

83  270 

83  276 

83  283 

83  289 

83  296 

83  302 

83  308 

681 

83  31S 

83  321 

83  327 

83  334 

83  340 

83  347 

83  353 

83  359 

83  366 

83  372 

682 

83378 

83  385 

83  391 

83  398 

83  404 

83  410 

83  417 

83  423 

83  429 

83  436 

683 

83  442 

83  448 

83  455 

83  461 

83  467 

83  474 

83  480 

83  487 

83  493 

83  499 

684 

83  506 

83  512 

83  518 

83  52s 

83  531 

83  537 

83  544 

83  550 

83  5S6 

83  563 

68s 

83  569 

83  575 

83  582 

83  588 

83  594 

83  601 

83  607 

83  613 

83  620 

83  626 

686 

83  632 

83639 

83  645 

83651 

83  6s8 

83  664 

83  670 

83  677 

83  683 

83  689 

687 

83  696 

83  702 

83  708 

83  715 

83  721 

83  727 

83  734 

83  740 

83  746 

83  753 

688 

83  759 

83  765 

83  771 

83  778 

83  784 

83  790 

83  797 

83  803 

83  809 

83  816 

689 

83  822 

83  828 

83  835 

83  841 

83  847 

83  853 

83  860 

83  866 

83  872 

83  879 

690 

83  885 

83  891 

83  897 

83  904 

83  910 

83  916 

83  923 

83  929 

83  935 

83  942 

691 

83  948 

83  954 

83  960 

83  967 

83  973 

83  979 

83  98s 

83  992 

83  998 

84  004 

692 

84  on 

84  017 

84  023 

84  029 

84  036 

84  042 

84  048 

84  OSS 

84  061 

84  067 

693 

84  073 

84  080 

84  086 

84  092 

84  098 

84  105 

84  III 

84  117 

84  123 

84  130 

694 

84  136 

84  142 

84  148 

84  155 

84  161 

84  167 

84  173 

84  180 

84  186 

84  192 

69s 

84  198 

84  20s 

84  211 

84  217 

84  223 

84  230 

84  236 

84  242 

84  248 

84  2SS 

696 

84  261 

84  267 

84  273 

84  280 

84  286 

84  292 

84  298 

84  30s 

84  311 

84  317 

697 

84323 

84  330 

84336 

84342 

84348 

84  354 

84361 

84367 

84  373 

84  379 

698 

84  386 

84  392 

84398 

84  404 

84  410 

84417 

84423 

84  429 

84435 

84442 

699 

84  448 

84454 

84  460 

84  466 

84  473 

84  479 

84  48s 

84  491 

84497 

84  504 

700 

84  sio 

84  S16 

84  522 

84  528 

84  535 

84  541 

84  547 

84  553 

84  559 

84  566 

701 

84  572 

84  578 

84  584 

84  590 

84  597 

84  603 

84  609 

84  61S 

84  621 

84  628 

702 

84  634 

84  640 

84  646 

84  652 

84  658 

8466s 

84  671 

84677 

84  683 

84  689 

703 

84  696 

84  702 

84  708 

84  714 

84  720 

84  726 

84  733 

84  739 

84  745 

84  751 

704 

84  757 

84  763 

84  770 

84  776 

84  782 

84  788 

84  794 

84  800 

84  807 

84  813 

70s 

84  819 

84  82s 

84831 

84  837 

84  844 

84  850 

84  8s6 

84  862 

84  868 

84  874 

706 

84  880 

84  887 

84  893 

84  899 

84  905 

84  911 

84  917 

84  924 

84  930 

84936 

707 

84  942 

84948 

84954 

84  960 

84  967 

84  973 

84  979 

84  98s 

84  991 

84  997 

708 

8S  003 

85  009 

8s  016 

8s  022 

8s  028 

85  034 

8s  040 

8s  046 

85  OS2 

85  0S8 

709 

85  06s 

85  071 

85  077 

85  083 

8s  089 

85  095 

85  lOI 

8s  107 

8s  114 

8s  120 

710 

85  126 

85  132 

85  138 

85  144 

8S  150 

8S  156 

85  163 

8s  169 

85  175 

8s  i8l 

711 

85  187 

85  193 

85  199 

85  205 

85  211 

85  217 

85  224 

8S  230 

8s  236 

85  242 

712 

8s  248 

85  254 

8s  260 

8s  266 

85  272 

85  278 

85  28s 

85  291 

85  297 

85  303 

713 

8S  309 

8S  315 

8s  321 

85  327 

85  333 

85  339 

8s  345 

85  352 

85  358 

8S  364 

714 

85  370 

85  376 

8s  382 

85  388 

85  394 

85  400 

85  406 

85  412 

8s  418 

8S  42s 

71S 

8S  431 

85  437 

85  443 

85  449 

85  455 

8s  461 

85  467 

85  473 

85  479 

85485 

716 

85  491 

85  497 

85  503 

85  509 

85  516 

85  522 

85  528 

85  534 

85  540 

8S  546 

717 

85  55  2 

85  558 

85  564 

85  570 

85  576 

85  582 

85  588 

85  594 

85  600 

85  606 

718 

85  612 

8s  618 

85625 

85  631 

85  637 

85  643 

85  649 

85  6SS 

8s  661 

8s  667 

719 

8S  673 

8S  679 

85  68s 

85  691 

85  697 

8S  703 

85  709 

85  715 

85  721 

85  727 

LOGARITHMS?  OF   NUMBERS 


247 


No. 

0 

I 

2 

3 

4 

5 

6 

7 

8 

9 

7J0 

85  733 

85  739 

85  745 

85  751 

8s  757 

85  763 

8s  769 

85  775 

85  78r 

8s  788 

721 

85  794 

8s  800 

8s  806 

85  812 

8s  818 

85  824 

85  830 

8s  836 

85  842 

85  848 

722 

85  8S4 

8s  860 

85  866 

8s  872 

85  878 

8s  884 

8s  890 

8s  896 

85  902 

85  908 

723 

85  914 

85  920 

85  926 

85  932 

85  938 

85  944 

85  950 

85  956 

8s  962 

8s  968 

724 

8S  974 

8s  980 

8s  986 

8s  992 

8s  998 

86  004 

86  010 

86  016 

86  022 

86  028 

72s 

86  034 

86  040 

86  046 

86  052 

86  058 

86  064 

86  070 

86  076 

86  082 

86088 

726 

86  094 

86  100 

86  106 

86  112 

86  118 

86  124 

86  130 

86  136 

86  141 

86  147 

727 

86  IS3 

86  IS9 

86  165 

86  171 

86  177 

86  183 

86  189 

86  195 

86  201 

86  207 

728 

86  213 

86  219 

86  225 

86  231 

86  237 

86  243 

86  249 

86  255 

86  261 

86  267 

729 

86  273 

86  279 

86  28s 

86  291 

86  297 

86  303 

86  308 

86314 

86  320 

86326 

730 

86  332 

86  338 

86344 

86  350 

86356 

86362 

86368 

86  374 

86  380 

86386 

731 

86  392 

86  398 

86  404 

86  410 

86  415 

86  421 

86  427 

86  433 

86  439 

8644s 

732 

86451 

86457 

86  463 

86  469 

86475 

86  481 

86  487 

86  493 

86  499 

86  504 

733 

86  sio 

86  516 

86  522 

86  528 

86  534 

86  540 

86  546 

86  552 

86558 

86  564 

734 

86  570 

86  576 

86  581 

86  587 

86  593 

86  599 

86  60s 

86  611 

86  617 

86623 

73S 

86  629 

86635 

86  641 

86  646 

86652 

86658 

86  664 

86  670 

86676 

86682 

736 

86  688 

86  694 

86  700 

86  70s 

86  711 

86717 

86  723 

86  729 

86  735 

86  741 

737 

86  747 

86  753 

86  759 

86  764 

86  770 

86  776 

86  782 

86  788 

86  794 

86  800 

738 

86  806 

86  812 

86  817 

86  823 

86  829 

86  83s 

86  841 

86  847 

86  853 

86  859 

739 

86  864 

86  870 

86  876 

86  882 

86  888 

86  894 

86  900 

86  906 

86  911 

86  917 

740 

86  923 

86  929 

86935 

86  941 

86  947 

86953 

86  958 

86  964 

86  970 

86  976 

741 

86  982 

86  988 

86  994 

86  999 

87  005 

87  on 

87  017 

87  023 

87  029 

87  03S 

742 

87  040 

87  046 

87  052 

87  058 

87  064 

87  070 

87  07s 

87  081 

87  087 

87  093 

743 

87  099 

87  105 

87  III 

87  116 

87  122 

87  128 

87  134 

87  140 

87  146 

87  151 

744 

87  157 

87  163 

87  169 

87  175 

87  181 

87  186 

87  192 

87  198 

87  204 

87  210 

745 

87  216 

87  221 

87  227 

87  233 

87  239 

87  24s 

87  251 

87  2S6 

87  262 

87  268 

746 

87  274 

87  280 

87  286 

87  291 

87  297 

87  303 

87  309 

87315 

87320 

87  326 

747 

87  332 

87  338 

87  344 

87  349 

87  355 

87  361 

87367 

87  373 

87  379 

87  384 

748 

87  390 

87  396 

87  402 

87  408 

87  413 

87  419 

87  42s 

87  431 

87  437 

87  442 

749 

87  448 

87  454 

87  460 

87  466 

87  471 

87  477 

87  483 

87  489 

87  495 

87  soo 

750 

87  506 

87  512 

87  S18 

87  523 

87  529 

87  535 

87  541 

87  547 

87  552 

87  5S8 

751 

87  564 

87  570 

87  576 

87  S8i 

87  587 

87  593 

87  599 

87  604 

87  610 

87  616 

752 

87  622 

87  628 

87633 

87  639 

8764s 

87  651 

87  656 

87  662 

87  668 

87  674 

753 

87  679 

87  685 

87  691 

87  697 

87  703 

87  708 

87  714 

87  720 

87  726 

87  731 

754 

87  737 

87  743 

87  749 

87  754 

87  760 

87  766 

87  772 

87  777 

87  783 

87  789 

755 

87  795 

87  800 

87  806 

87  812 

87  818 

87  823 

87  829 

87  83s 

87  841 

87  846 

756 

87  852 

87  858 

87  864 

87  869 

87  875 

87  881 

87  887 

87  892 

87  898 

87  904 

757 

87  910 

87  915 

87  921 

87  927 

87  933 

87  938 

87  944 

87  950 

87  955 

87  961 

7S8 

87  967 

87  973 

87  978 

87  984 

87  990 

87  996 

88  001 

88  007 

88  013 

88  018 

759 

88  024 

88  030 

88  036 

88  041 

88  047 

88  053 

88  058 

88  064 

88  070 

88  076 

760 

88  081 

88  087 

88  093 

88  098 

88  104 

88  no 

88  116 

88  121 

88  127 

88  133 

761 

88  138 

88  144 

88  150 

88  is6 

88  161 

88  167 

88  173 

88  178 

88  184 

88  190 

762 

88  195 

88  201 

88  207 

88  213 

88  218 

88  224 

88  230 

88235 

88  241 

88  247 

763 

88  252 

88  258 

88  264 

88  270 

88  275 

88  281 

88  287 

88  292 

88  298 

88  304 

764 

88  309 

88315 

88  321 

88  326 

88  3i2 

88  338 

88  343 

88  349 

88355 

88  3O0 

248 


APPENDIX 


No. 

0 

I 

2 

3 

4 

5 

6 

7 

8 

9 

76s 

88  366 

88  372 

88  377 

88383 

88  389 

88  395 

88  400 

88  406 

88  412 

88  417 

766 

88  423 

88  429 

88434 

88  440 

88  446 

88  451 

88  457 

88  463 

88  468 

88  474 

767 

88  480 

88  48s 

88  491 

88  497 

88  502 

88  508 

88  S13 

88  519 

88  525 

88  530 

768 

88  536 

88  542 

88  S47 

88  553 

88  559 

88  564 

88  570 

88  576 

88  581 

88  587 

769 

88  593 

88  598 

88  604 

88  610 

88  61S 

88  621 

88  627 

88  632 

88  638 

88  643 

770 

88  649 

88  655 

88  660 

88  666 

88  672 

88  677 

88  683 

88  689 

88  694 

88  700 

771 

88  705 

88  711 

88  717 

88  722 

88  728 

88  734 

88  739 

88  745 

88  750 

88  756 

772 

88  762 

88  767 

88773 

88  779 

88  784 

88  790 

88  795 

88  801 

88  807 

88  812 

773 

88  818 

88  824 

88  829 

88  835 

88  840 

88  846 

88  852 

88  857 

88  863 

88  868 

774 

88  874 

88  880 

88  88s 

88  891 

88  897 

88  902 

88  908 

88  913 

88  919 

88  925 

775 

88  930 

88  936 

88  941 

88  947 

88  953 

88  958 

88  964 

88  969 

88  975 

38  981 

776 

88  986 

88  992 

88  997 

89  003 

89  009 

89  014 

89  020 

89  02s 

89  031 

89  037 

Tn 

89  042 

89  048 

89  053 

89  059 

89  064 

89  070 

89  076 

89  081 

89  087 

89  092 

778 

89  098 

89  104 

89  109 

89  115 

89  120 

89  126 

89  131 

89  137 

89  143 

89  148 

779 

89  154 

89  159 

89  165 

89  170 

89  176 

89  182 

89  187 

89  193 

89  198 

89  204 

780 

89  209 

89  215 

89  221 

89  226 

89  232 

89  237 

89  243 

89  248 

89  254 

89  260 

781 

89  265 

89  271 

89  276 

89  282 

89  287 

89  293 

89  298 

89  304 

89  310 

89  315 

782 

89  321 

89326 

89332 

89  337 

89  343 

89348 

89  354 

89  360 

89365 

89  371 

783 

89376 

89382 

89387 

89393 

89  398 

89  404 

89  409 

89  41S 

89  421 

89  426 

784 

89  432 

89437 

89443 

89  448 

89  454 

89459 

89  46s 

89  470 

89476 

89  481 

78s 

89487 

89  492 

89  498 

89  S04 

89  509 

89  515 

89  520 

89  526 

89  531 

89  S37 

786 

89  542 

89548 

89  553 

89  559 

89  564 

89  570 

89  575 

89  581 

89  586 

89  592 

787 

89  597 

89  603 

89  609 

89  614 

89  620 

89625 

89  631 

89  636 

89  642 

89647 

788 

89653 

89  658 

89  664 

89  669 

89675 

89  680 

89  686 

89  691 

89  697 

89  702 

789 

89  708 

89  713 

89  719 

89  724 

89  730 

89  735 

89  741 

89  746 

89  752 

89  757 

790 

89  763 

89  768 

89  774 

89  779 

89  785 

89  790 

89  706 

89  8or 

89  807 

89  812 

791 

89  818 

89  823 

89  829 

89  834 

89  840 

89  845 

89  851 

89  856 

89  862 

89  867 

792 

89873 

89  878 

89  883 

89  889 

89  894 

89  900 

89  905 

89  911 

89  916 

89  922 

793 

89  927 

89  933 

89  938 

89  944 

89  949 

89  955 

89  960 

89  966 

89  971 

89  977 

794 

89  982 

89  988 

89  993 

89  998 

90  004 

90  009 

90  ors 

90  020 

90  026 

90  031 

795 

90  037 

90  042 

90  048 

90  053 

90  059 

90  064 

90  069 

90  075 

90  080 

90  086 

796 

90  091 

90097 

90  102 

90  108 

90  113 

90  1 19 

90  124 

90  129 

90  135 

90  140 

797 

90  146 

90  151 

90  157 

90  162 

90  168 

90  173 

90  179 

90  184 

90  189 

90  195 

798 

90  200 

90  206 

90  211 

90  217 

90  222 

90  227 

90  233 

90  238 

90  244 

90  249 

709 

90  255 

90  260 

90  266 

90  271 

90  276 

90  282 

90  287 

90  293 

90  298 

90  304 

800 

90  309 

90  314 

90  320 

90  325 

90  a  I 

90336 

90  342 

90  347 

90  352 

90  3S8 

801 

90  363 

90369 

90374 

90  380 

90  385 

90  390 

90  396 

90  401 

90  407 

90  412 

802 

90  417 

90423 

90  428 

90  434 

90  439 

9044s 

90  450 

90  455 

90  461 

90  466 

803 

90  472 

90477 

90  482 

90488 

90493 

90  499 

90  504 

90  509 

90  515 

90  520 

804 

90  526 

90  531 

90  536 

90  542 

90  547 

90  553 

90  558 

90  563 

90  S69 

90S74 

80s 

90  580 

90  58s 

90  590 

90  596 

90  601 

90  607 

90  612 

90  617 

90  623 

90628 

806 

90634 

90  639 

9C644 

90650 

90655 

90  660 

90  666 

90  671 

90  677 

90  682 

807 

90  687 

90  693 

90  698 

90  703 

90  709 

90  714 

90  720 

90  725 

90  730 

90  736 

808 

90  741 

90  747 

90  752 

90  757 

90  763 

90  768 

90  773 

90  779 

90  784 

90  789 

809 

90  795 

90  800 

90  806 

90  8  n 

90  816 

90  822 

90  827 

90  832 

90  838 

90  843 

LOGARITHMS  OF   NUMBERS 


249 


No. 

0 

I 

2 

3 

4 

5 

6 

7 

8 

9 

810 

90  849 

90  854 

90  859 

90  86s 

90  870 

90  875 

90  881 

90  886 

90  891 

90  897 

811 

90  902 

90907 

90  913 

90  918 

90  924 

90  929 

90  934 

90  940 

90  945 

90950 

812 

90  956 

90  961 

90  966 

90  972 

90  977 

90  982 

90  988 

90993 

90  998 

91  004 

813 

91  009 

91  014 

91  020 

91  025 

91  030 

91  036 

91  041 

91  046 

91  052 

91  057 

814 

91  062 

91  068 

91  073 

91  078 

91  084 

91  089 

91  094 

91  100 

91  105 

91  no 

81S 

91  116 

91  121 

91  126 

91  132 

91  137 

91  142 

91  148 

91  153 

91  158 

91  164 

816 

91  169 

91  174 

91  180 

91  185 

91  190 

91  196 

91  201 

91  206 

91  212 

91  217 

817 

9t  222 

91  228 

91  233 

91  238 

91  243 

91  249 

91  254 

91  259 

91  265 

91  270 

818 

91  275 

91  281 

91  286 

91  291 

91  297 

91  302 

91  307 

91  312 

91  318 

91  323 

819 

91  328 

91  334 

91  339 

91  344 

91  350 

91  355 

91  360 

91  365 

91  371 

91  376 

820 

91  381 

91  387 

91  392 

91  397 

91  403 

91  408 

91  413 

91  418 

91  424 

91  429 

821 

91  434 

91  440 

91  445 

91  450 

91  455 

91  461 

91  466 

91  471 

91  477 

91  482 

822 

91  487 

91  492 

91  498 

91  S03 

91  508 

91  514 

91  519 

91  524 

91  529 

91  535 

823 

91  540 

91  545 

91  551 

91  5S6 

91  561 

91  566 

91  572 

91  577 

91  582 

91  587 

824 

91  593 

91  598 

91  603 

91  609 

91  614 

91  619 

91  624 

91  630 

91  635 

91  640 

82s 

91  64s 

91  651 

91  656 

91  661 

91  666 

91  672 

91  677 

91  682 

91  687 

91  693 

826 

91  698 

91  703 

91  709 

91  714 

91  719 

91  724 

91  730 

91  735 

91  740 

91  745 

827 

91  751 

91  756 

91  761 

91  766 

91  772 

91  777 

91  782 

91  787 

91  793 

91  798 

828 

91  803 

91  808 

91  814 

91  819 

91  824 

91  829 

91  834 

91  840 

91  84s 

91  850 

829 

91  8S5 

91  861 

91  866 

91  871 

91  876 

91  882 

91  887 

91  892 

91  897 

91  903 

830 

91  908 

91  913 

91  918 

91  924 

91  929 

91  934 

91  939 

91  944 

91  950 

91  955 

831 

91  960 

91  965 

91  971 

91  976 

91  981 

91  986 

91  991 

91  997 

92  002 

92  007 

832 

92  012 

92  018 

92  023 

92  028 

92  033 

92  038 

92  044 

92  049 

92  054 

92  059 

833 

92  06s 

92  070 

92  075 

92  080 

92  08s 

92  091 

92  096 

92  lOI 

92  106 

92  III 

834 

92  117 

92  122 

92  127 

92  132 

92  137 

92  143 

92  148 

92  153 

92  158 

92  163 

83  s 

92  169 

92  174 

92  179 

92  184 

92  189 

92  195 

92  200 

92  205 

92  210 

92  21S 

836 

92  221 

92  226 

92  231 

92  236 

92  241 

92  247 

92  252 

92  257 

92  262 

92  267 

837 

92  273 

92  278 

92  283 

92  288 

92  293 

92  298 

92  304 

92  309 

92  314 

92319 

838 

92  324 

92  330 

92  335 

92  340 

92  345 

92  350 

92  355 

92  361 

92  366 

92  371 

839 

92  376 

92  381 

92  387 

92  392 

92  397 

92  402 

92  407 

92  412 

92  418 

92  423 

840 

92  428 

92  433 

92  438 

92  443 

92  449 

92  454 

92  459 

92  464 

92  469 

92  474 

841 

92  480 

92  485 

92  490 

92  495 

92  500 

92  505 

92  511 

92  516 

92  521 

92  526 

842 

92  531 

92  536 

92  542 

92  547 

92  552 

92  557 

92  562 

92  567 

92  572 

92  578 

843 

92  583 

92  588 

92  593 

92  598 

92  603 

92  609 

92  614 

92  619 

92  624 

92  629 

844 

92  634 

92  639 

92  645 

92  650 

92  655 

92  660 

92  665 

92  670 

92  67s 

92  681 

84s 

92  686 

92  691 

92  696 

92  701 

92  706 

92  711 

92  716 

92  722 

92  727 

92  732 

846 

92  737 

92  742 

92  747 

92  752 

92  758 

92  763 

92  768 

92  773 

92  778 

92  783 

847 

92  788 

92  793 

92  799 

92  804 

92  809 

92  814 

92  819 

92  824 

92  829 

92  834 

848 

92  840 

92  845 

92  850 

92  8S5 

92  860 

92  86s 

92  870 

92  87s 

92  881 

92  886 

849 

92  891 

92  896 

92  901 

92  906 

92  911 

92  916 

92  921 

92  927 

92  932 

92  937 

850 

92  942 

92  947 

92  952 

92  957 

92  962 

92  967 

92  973 

92  978 

92  983 

92  988 

851 

92  993 

92  998 

93  003 

93  008 

93  013 

93  018 

93  024 

93  029 

93  034 

93  039 

852 

93  044 

93  049 

93  054 

93  059 

93  064 

93  069 

93  075 

93  080 

93  08s 

93  090 

853 

93  095 

93   100 

93  105 

93  no 

93  115 

93  120 

93  125 

93  131 

93   136 

93  141 

854 

93  146 

93  151 

93  156 

93  161 

93  166 

93   171 

93  176 

93  181 

93  186 

93  192 

250 


APPENDIX 


No. 

0 

I 

2 

3 

4 

5 

6 

7 

8 

9 

85S 

93  197 

93  202 

93  207 

93  212 

93  217 

93  222 

93  227 

93  232 

93  237 

93  242 

8s6 

93  247 

93  252 

93  258 

93  263 

93  268 

93  273 

93  278 

93  283 

93  288 

93  293 

857 

93  298 

93  303 

93  308 

93  313 

93  318 

93  323 

93  328 

93  334 

93  339 

93  344 

858 

93  349 

93  354 

93  359 

93  364 

93  369 

93  374 

93  379 

93  384 

93  389 

93  394 

859 

93  399 

93  404 

93  409 

93  414 

93  420 

93  425 

93  430 

93  435 

93  440 

93  445 

860 

93  450 

93  455 

93  460 

93  465 

93  470 

93  475 

93  480 

93  485 

93  490 

93  495 

861 

93  500 

93  SOS 

93  510 

93  515 

93  520 

93  526 

93  S3  I 

93  536 

93  541 

93  546 

862 

93  551 

93  556 

93  561 

93  566 

93  571 

93  576 

93  581 

93  586 

93  591 

93  596 

863 

93  601 

93  606 

93  611 

93  616 

93  621 

93  626 

93  631 

93  636 

93  641 

93  646 

864 

93  651 

93  656 

93  661 

93  666 

93  671 

93  676 

93  682 

93  687 

93  692 

93  697 

86s 

93  702 

93  707 

93   712 

93  717 

93  722 

93  727 

93  732 

93  737 

93  742 

93  747 

866 

93  752 

93  757 

93  762 

93  767 

93  772 

93  777 

93  782 

93   787 

93  792 

93  797 

867 

93  802 

93  807 

93  812 

93  817 

93  822 

93  827 

93  832 

93   837 

93  842 

93  847 

868 

93  852 

93  8s  7 

93  862 

93  867 

93  872 

93  877 

93  882 

93  887 

93  892 

93  897 

869 

93  902 

93  907 

93  912 

93   917 

93  922 

93  927 

93  932 

93  937 

93  942 

93  947 

870 

93  952 

93  957 

93  962 

93  967 

93  972 

93  977 

93  982 

93  987 

93  992 

93  997 

871 

94  002 

94  007 

94  012 

94017 

94  022 

94  027 

94  032 

94  037 

94  042 

94  047 

872 

94  052 

94  057 

94  062 

94  067 

94  072 

94  077 

94  082 

94  086 

94  091 

94  096 

873 

94  lOl 

94  106 

94  III 

94  116 

94  121 

94  126 

94  131 

94  136 

94  141 

94  146 

874 

94  151 

94  156 

94  161 

94  166 

94  171 

94  176 

94  181 

94  186 

94  191 

94  196 

875 

94  201 

94  206 

94  211 

94  216 

94  221 

94  226 

94  231 

94  236 

94  240 

94  24s 

876 

94  250 

94  255 

94  260 

94  26s 

94  270 

94  275 

94  280 

94  28s 

94  290 

94  295 

877 

94  300 

94  30s 

94  310 

9431S 

94  320 

94  32s 

94  330 

94  335 

94  340 

94  345 

878 

94  349 

94  354 

94  3S9 

94  364 

94  369 

94  374 

94  379 

94  384 

94  389 

94  394 

879 

94  399 

94  404 

94  409 

94  414 

94  419 

94424 

94  429 

94  433 

94  438 

94  443 

880 

94  448 

94  453 

94  458 

94  463 

94  468 

94  473 

94  478 

94  483 

94488 

94  493 

881 

94  498 

94  503 

94  507 

94  512 

94  S17 

94  522 

94  527 

94  532 

94  537 

94  542 

882 

94  547 

94  552 

94  557 

94  562 

94  567 

94571 

94  576 

94  S8i 

94  586 

94  591 

883 

94  596 

94  601 

94  606 

94  611 

94  6i6 

94  621 

94  626 

94  630 

94  635 

94  640 

884 

94  645 

94  650 

94  655 

94  660 

94  665 

94  670 

94  675 

94  680 

94  685 

94  689 

88s 

94  694 

94  699 

94  704 

94  709 

94  714 

94  719 

94  724 

94  729 

94  734 

94  738 

886 

94  743 

94  748 

94  753 

94  758 

94  763 

94  768 

94  773 

94  778 

94  783 

94  787 

887 

94  792 

94  797 

94  802 

94  807 

94  812 

94  817 

94  822 

94  827 

94  832 

94  836 

888 

94  841 

94  846 

94  851 

94  856 

94  861 

94  866 

94  871 

94  876 

94  880 

94  885 

889 

94  890 

94  895 

94  900 

94  905 

94  910 

94  91S 

94  919 

94  924 

94  929 

94  934 

890 

94  939 

94  944 

94  949 

94  954 

94  959 

94  963 

94  968 

94  973 

94978 

94  983 

891 

94  988 

94  993 

94  998 

95  002 

95  007 

95  012 

95  017 

95  022 

95  027 

95  032 

892 

95  036 

95  041 

95  046 

95  OS  I 

95  056 

95  061 

95  066 

95  071 

95  075 

95  080 

893 

95  085 

95  090 

95  095 

95  100 

95  105 

95  109 

95  114 

95  119 

95  124 

95  129 

894 

95  134 

95  139 

95  143 

95  148 

95  153 

95  158 

95  163 

95  168 

95  173 

95  177 

89s 

95  182 

95  187 

95  192 

95  197 

95  202 

95  207 

95  211 

95  216 

95  221 

95  226 

896 

95  231 

95  236 

95  240 

95  245 

95  250 

95  255 

95  260 

95  265 

95  270 

95  274 

897 

95  279 

95  284 

95  289 

95  294 

95  299 

95  303 

95  308 

95  313 

95  318 

95  323 

898 

95  328 

95  332 

95  337 

95  342 

95  347 

95  352 

95  357 

95  361 

95  366 

95  371 

899 

95  376 

95  381 

95  386 

95  390 

95  395 

95  400 

95  405 

95  410 

95  415 

95  419 

LOGARITHMS  OF  NUMBERS 


251 


No. 

0 

I 

2 

3 

4 

5 

6 

7 

8 

9 

900 

95  424 

95  429 

95  434 

95  439 

95  444 

95  448 

95  453 

95  458 

95  463 

95  468 

901 

95  472 

95  477 

95  482 

95  487 

95  492 

95  497 

95  501 

95  506 

95  511 

95  516 

902 

95  521 

95  52s 

95  530 

95  535 

95  540 

95  545 

95  550 

95  554 

95  559 

95  564 

903 

95  569 

95  574 

95  578 

95  583 

95  588 

95  593 

95  598 

95  602 

95  607 

95  612 

904 

95617 

95  622 

95  626 

95  631 

95  636 

95  641 

95  646 

95  650 

95  6s5 

95  660 

90s 

95665 

95  670 

95  674 

95  679 

95  684 

95  689 

95  694 

95  698 

95  703 

95  708 

906 

95  713 

95  718 

95  722 

95  727 

95  732 

95  737 

95  742 

95  746 

95  751 

95  756 

907 

95  761 

95  766 

95  770 

95  775 

95  780 

95  785 

95  789 

95  794 

95  799 

95  804 

908 

95  809 

95  813 

95  818 

95  823 

95  828 

95  832 

95  837 

95  842 

95  847 

95  852 

909 

95  856 

9S  861 

95  866 

95  871 

95  875 

95  880 

95  885 

95  890 

95  895 

95  899 

910 

95  904 

95  909 

95  914 

95  918 

95  923 

95  928 

95  933 

95  938 

95  942 

95  947 

911 

95  952 

95  957 

95  961 

95  966 

95  971 

95  976 

95  980 

95  985 

95  990 

95  995 

912 

95  999 

96  004 

96  009 

96  014 

96  019 

96  023 

96  028 

96  033 

96  038 

96  042 

913 

96  047 

96  052 

96  057 

96  061 

96  066 

96  071 

96  076 

96  080 

96  085 

96  090 

914 

96  095 

96  099 

96  104 

96  109 

96  114 

96  118 

96  123 

96  128 

96  133 

96  137 

91S 

96  142 

96  147 

96  152 

96  156 

96  161 

96  166 

96  171 

96  175 

96  180 

96  185 

916 

96  190 

96  194 

96  199 

96  204 

96  209 

96  213 

96  218 

96  223 

96  227 

96  232 

917 

96  237 

96  242 

96  246 

96  25  I 

96  256 

96  261 

96  265 

96  270 

96  275 

96  280 

918 

96  284 

96  289 

96  294 

96  298 

96  303 

96  308 

96  313 

96  317 

96  322 

96  327 

919 

96  332 

96  336 

96  341 

96  346 

96  350 

96  355 

96  360 

96  365 

96  369 

96374 

920 

96  379 

96  384 

96  388 

96  393 

96  398 

96  402 

96  407 

96  412 

96  417 

96  421 

921 

96  426 

96  431 

96  435 

96  440 

96  445 

96  450 

96  454 

96  459 

96  464 

96  468 

922 

96  473 

96  478 

96  483 

96  487 

96  492 

96  497 

96  501 

96  506 

96  511 

96  515 

923 

96  520 

96  525 

96  530 

96  534 

96  539 

96  544 

96  548 

96  553 

96  558 

96  562 

924 

96  567 

96  S72 

96  577 

96  S8l 

96  586 

96  591 

96  595 

96  600 

96  605 

96  609 

92s 

96  614 

96  619 

96  624 

96  628 

96  633 

96  638 

96  642 

96  647 

96  652 

96  656 

926 

96  661 

96  666 

96  670 

96  675 

96  680 

96  685 

96  689 

96  694 

96  699 

96  703 

927 

96  708 

96  713 

96  717 

96  722 

96  727 

96  731 

96  736 

96  741 

96  745 

96  750 

928 

96  755 

96  759 

96  764 

96  769 

96  774 

96  778 

96  783 

96  788 

96  792 

96  797 

929 

96  802 

96  806 

96  811 

96  816 

96  820 

96  825 

96  830 

96  834 

96  839 

96  844 

930 

96  848 

96  853 

96  858 

96  862 

96  867 

96  872 

96  876 

96  881 

96  886 

96  890 

931 

96  895 

96  900 

96  904 

96  909 

96  914 

96  918 

96  923 

96  928 

96  932 

96  937 

932 

96  942 

96  946 

96  951 

96  956 

96  960 

96  965 

96  970 

96  974 

96  979 

96  984 

933 

96  988 

96  993 

96  997 

97  002 

97  007 

97  on 

97  016 

97  021 

97  025 

97  030 

934 

97  035 

97  039 

97  044 

97  049 

97  053 

97  058 

97  063 

97  067 

97  072 

97  077 

935 

97  081 

97  086 

97  090 

97  095 

97  100 

97  104 

97  109 

97  114 

97  118 

97  123 

936 

97  128 

97  132 

97  137 

97  142 

97  146 

97  151 

97  155 

97  160 

97  165 

97  169 

937 

97  174 

97  179 

97  183 

97  188 

97  192 

97  197 

97  202 

97  206 

97  211 

97  216 

938 

97  220 

97  225 

97  230 

97  234 

97  239 

97  243 

97  248 

97  253 

97  257 

97  262 

939 

97  267 

97  271 

97  276 

97  280 

97  285 

97  290 

97  294 

97  299 

97  304 

97  308 

940 

97313 

97  317 

97  322 

97  327 

97  331 

97  336 

97  340 

97  345 

97  350 

97  354 

941 

97  3S9 

97  364 

97  368 

97  373 

97  377 

97  382 

97387 

97  391 

97  396 

97  400 

942 

97  405 

97  410 

97  414 

97  419 

97  424 

97  428 

97  433 

97  437 

97  442 

97  447 

943 

97  45 1 

97  456 

97  460 

97  46s 

97  470 

97  474 

97  479 

97  483 

97  488 

97  493 

944 

97  497 

97  502 

97  506 

97  511 

97  516 

97  520 

97  525 

97  529 

97  534 

97  539 

252 


APPENDIX 


No. 

0 

I 

2 

3 

4 

5 

6 

7 

8 

9 

945 

97  543 

97  548 

97  552 

97  557 

97  562 

97  566 

97  571 

97  575 

97  580 

97  585 

946 

97  589 

97  594 

97  598 

97  603 

97  607 

97  612 

97  617 

97  621 

97  626 

97  630 

947 

97  635 

97  640 

97  644 

97  649 

97  653 

97  658 

97  663 

97  667 

97  672 

97  676 

948 

97  681 

97  685 

97  690 

97  695 

97  699 

97  704 

97  708 

97  713 

97  717 

97  722 

949 

97  727 

97  731 

97  736 

97  740 

97  745 

97  749 

97  754 

97  759 

97  763 

9776C 

950 

97  772 

97  777 

97  782 

97  786 

97  791 

97  795 

97  800 

97  804 

97  809 

97  813 

951 

97  818 

97  823 

97  827 

97  832 

97  836 

97  841 

97  845 

97  850 

97  855 

97  859 

952 

97  864 

97  868 

97  873 

97  877 

97  882 

97  886 

97  891 

97  896 

97  900 

97  905 

953 

97  909 

97  914 

97  918 

97  923 

97  928 

97  932 

97  937 

97  941 

97  946 

97  950 

9S4 

97  955 

97  959 

97  964 

97  968 

97  973 

97  978 

97  982 

97  987 

97  991 

97  996 

955 

98  000 

98  005 

98  009 

98  014 

98  019 

98  023 

98  028 

98  032 

98  037 

98  041 

956 

98  046 

98  050 

98  055 

98  059 

98  064 

98  068 

98  073 

98  078 

98  082 

98  087 

957 

98  091 

98  096 

98  100 

98  105 

98  109 

98  114 

98  118 

98  123 

98  127 

98  132 

958 

98  137 

98  141 

98  146 

98  150 

98  155 

98  159 

98  164 

98  168 

98  173 

98  177 

959 

98  182 

98  186 

98  191 

98  195 

98  200 

98  204 

98  209 

98  214 

98  218 

98  223 

960 

98  227 

98  232 

98  236 

98  241 

98  245 

98  250 

98  254 

98  259 

98  263 

98  268 

961 

98  272 

08  277 

98  281 

98  286 

98  290 

98  295 

98  299 

98  304 

98  308 

98  313 

962 

98  318 

98  322 

98  327 

98  331 

98  336 

98  340 

98  345 

98  349 

98354 

98358 

963 

98  363 

98367 

98  372 

98  376 

98  381 

98  38s 

98  390 

98  394 

98  399 

98  403 

964 

98  408 

98  412 

98  417 

98  421 

98  426 

98  430 

98  435 

98  439 

98  444 

98  448 

96s 

98  453 

98  457 

98  462 

98  466 

98  471 

98  475 

98  480 

98  484 

98  489 

98  493 

966 

98  498 

98  502 

98  507 

98  511 

98  516 

98  520 

98  525 

98  529 

98  534 

98  538 

967 

98  543 

98  547 

98  552 

98  556 

98  S6i 

98  56s 

98  570 

98  574 

98  579 

98  583 

968 

98  S88 

98  592 

98  597 

98  601 

98  605 

98  610 

98  614 

98  619 

98  623 

98  62S 

969 

98  632 

98  637 

98  641 

98  646 

98  650 

98  655 

98  659 

98  664 

98  668 

98673 

970 

98  677 

98  682 

98  686 

98  691 

98  695 

98  700 

98  704 

98  709 

98  713 

98  717 

971 

98  722 

98  726 

98  731 

98  735 

98  740 

98  744 

98  749 

98  753 

98  758 

98  762 

972 

98  767 

98  771 

98  776 

98  780 

98  784 

98  789 

98  793 

98  798 

98  802 

98  807 

973 

98  811 

98  816 

98  820 

98  82s 

98  829 

98  834 

98  838 

98  843 

98  847 

98  851 

974 

98  856 

98  860 

98  865 

98  869 

98  874 

98  878 

98  883 

98  887 

98  892 

98  896 

975 

98  900 

98  90s 

98  909 

98  914 

98  918 

98  923 

98  927 

98  932 

98  936 

98  941 

976 

98  945 

98  949 

98  954 

98  958 

98  963 

98  967 

98  972 

98  976 

98  981 

98985 

977 

98  989 

98  994 

98  998 

99  003 

99  007 

99  012 

99  016 

99  021 

99  025 

99  029 

978 

99  034 

99  038 

99  043 

99  047 

99  052 

99  056 

99  061 

99  065 

99  069 

99  074 

979 

99  078 

99  083 

99  087 

99  092 

99  096 

99  100 

99  105 

99  109 

99  114 

99  118 

980 

99  123 

09  127 

99  131 

99  136 

99  140 

99  145 

99  149 

99  154 

99  158 

99  162 

981 

99  167 

99  171 

99  176 

99  180 

99  185 

99  189 

99  193 

99  198 

99  202 

99  207 

982 

99  211 

99  216 

99  2  20 

99  224 

99  229 

99  233 

99  238 

99  242 

99  247 

99  251 

983 

99  255 

99  260 

99  264 

99  269 

99  273 

99  277 

99  282 

99  286 

99  291 

99  295 

984 

99  300 

99  304 

99308 

99  313 

99  317 

99  322 

99  326 

99  330 

99  335 

99  339 

98s 

99  344 

99  348 

99  352 

99  357 

99361 

99  366 

99  37c 

99  374 

99  379 

99383 

986 

99388 

99  392 

99  396 

99  401 

99  405 

99410 

99  414 

99  419 

99  423 

99427 

987 

99432 

99  436 

99  441 

99  445 

99  449 

99  454 

99  458 

99463 

99  467 

99  471 

988 

99  476 

99  480 

99  484 

99  489 

99  493 

99  498 

99  502 

99  506 

99  5  I  I 

99  SIS 

989 

99  5  20 

99  524 

99  528 

99  533 

99  537 

99  542 

99  546 

99  550 

99  555 

99  SS9 

LOGARITHMS  OP   NUMBERS 


253 


No. 

0 

I 

2 

3 

4 

5 

6 

7 

8 

9 

990 

99  564 

99  568 

99  572 

99  577 

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COMPOUND  INTEREST  AND  OTHER  COMPUTATIONS    257 


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258 


APPENDIX 


65 


Oi  r*    -^   t^  >0 

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INDEX 


Accounts,  averaging,  42-49 
Addition, 

fractions,  16 

short  methods,  4 
Amortization,  bonds,  194-216 
Annuities,  171-189 

logarithms,  190-193 

present  worth,  186 

tables,  262-269 
Arithmetical  progression,  22-28 
Averages,  29-41 

accounts,  42-49 

compound,  47-49 

dates,  42-45 

in  foreign  exchange,  128 

interest,  42 

moving,  30 

periodic,  37 

progressive,  34 

settling  an  account,  42 

simple,  29 

weighted,  38 


B 


Balancing,  short  methods,  3 
Banking,    clearing    house    problems, 

98-102 
Bonds, 

amortization,  194-216 

discount,  194-216 

income  rates,  201 

premiums,  203 

present  worth,  205 

price  computation,  205 


Building  and  loan  associations,  103- 

115 
Dayton  or  Ohio  plan,  1 12 
Dexter 's  rule,  108 
income  of,  no 
individual  plan,  1 1 1 
partnership  plan,  107 
premiums,  1 10 
serial  plan,  106 
terminating  plan,  103 
withdrawal  of  shares,  1 10 


Capital, 

capitalizing  gross  income,  118 
reducing  to  profit  and  loss  ratio,  95 
working, 

basis  of  turnover,  85 
defined,  87 

Cash  discount,  80-82 

Clearing  house,  98-102 

Complements,  8 

Consolidations,  1 16-122 

ascertaining  earning  power,  116 
capitalizing  gross  income,  118 
stock  issues,  116,  120 
valuation  of  good-will,  117 

Conversion, 

foreign  branch  accounts,  130 
foreign  exchange,  123-134 

Currency,  foreign,  table  of  values,  231 


Dates, 
averages,  42-45,  128 


271 


272 


INDEX 


Dates — Continued 

current    account    in    foreign     ex- 
change, 128 

reducing  to  months,  45 
Dayton     plan,     building    and     loan 

associations,  1 12 
Debts,  present  value,  182 
Decrease       (See      "Increase       and 

decrease") 
Depreciation,  221-230 
Dexter 's  rule,  108 
Discount  (See  also  "Amortization") 

cash,  80-82 

compound,  169 

trade,  76-79 
Division, 

logarithms,  137 

short  methods,  13 


E 


Income, 

capitalizing  gross,  118 

consolidations,  1 16-122 
Increase  and  decrease, 

averages,  32-37 

percentage,  51 

short  methods,  7 
Individual   plan,   building  and  loan 

association,  ii\ 
Interest,  42,  154-170 

logarithms,  190-193 

rents,  217-220 

tables,  254-269 
Interpolation,  logarithms,  150 
Inventory, 

gross    profit    method    of    approxi 
mating,  56 

turnover,  84 


Earnings,  ascertaining  of,  116 

Equations,  66-75 

Exchange  (See  "Foreign  exchange") 


Focal  date,  44 

Foreign     branches,     conversion     of 

accounts,  130 
Foreign  exchange,  123-134 

average  date  of  current  account, 

128 
table  of  values,  231 
Fractions,  15-20 


Gross  prof.t  method  approximating 

inventory,  56 
Good-will,  1 16-122 
valuation  of,  117 


Leaseholds,  217-220 
Liquidation,  of  partnership,  94 
Logarithms,  135-153 

annuities,  190-193 

characteristic,  140 
negative,  146 

compound  interest,  190-193 

division,  137 

interpolation,  150 

mantissa,  140 

multiplication,  136 

nature  of,  140 

powers,  137 

roots,  139 

tables,  233-253 

M 

Mantissa,  140 
Multiplication, 

fractions,  16 

logarithms,  136 

short  methods,  9-13 


INDEX 


273 


N 

Net  decrease,  short  method,  7 

O 

Ohio  plan,  building  and  loan  asso- 
ciations, 112 


Partial  payments, 

building  and  loan  associations,  103 

interest  calculations,  157 
Partnership,  89-97 

building  and  loan  associations,  197 

liquidation,  94 

periodical  distribution,  94 

profit  and  loss,  89-94 
Percentage,  50-65 

apportionment,  55   . 

inventory,  56 

sales,  51-55 

statements,  analysis  of,  60-65 
Powers,  logarithms,  137 
Premiums,   building  and   loan   asso- 
ciations, no 
Present  worth, 

annuities,  175 

annuities  by  logarithms,  190-193 

bonds,  205 

calculation  of,  167 

debts,  182 

rent,  179 

sinking  funds,  174,  185 

tables,  254-269 
Price,  trade  discount,  77 
Profit  and  loss, 

partnership,  89-94 

reducing  capital  to  ratio,  95 
Progressions,  arithmetical,  22-28 
Proofs,  20 

R 

Reciprocals,  division  by  use  of,  13 
Rent,  217-220 


Rent — Continued 

present  worth,  179 
Roots,  logarithms,  139 


Sales,  percentage,  51-55 
Serial  plan,  building  and  loan  asso- 
ciations, 106 
Shares     (See  "Stock") 
Short  methods, 

addition,  4 

balancing,  3 

division,  13 

increase  and  decrease,  7 

multiplication,  9-13 

subtraction,  6 
Sinking  funds,   present   worth,    174, 

185 
Stock, 

building    and     loan    associations, 

methods  of,  103-115 
issues  of,  in  consolidation,  116,  120 
Stock-in-trade     (See  "  Inventory ' ') 
Solutions,  equations,  66-75 
Statements,  percentage  analysis,  60- 

65 

Subtraction, 
fractions,  16 
short  methods,  6 


Tables, 

foreign  currency,  231 

interest,  254-269 

logarithms,  233-25,7, 
Terminal    plan,    building    and    loan 

associations,  103 
Trade  discount,  76-79 
Turnover,  83-88 

working  capital  as  basis,  85 


274 


INDEX 


Valuation, 

good- will,  117 
Values, 

foreign  currency,  231 

W 
Weighted  averages,  38 


Withdrawal  of  shares. 

Building    and    Loan    Associations, 
no 
Working  capital, 

basis  of  turnover,  85 

defined,  87 
Worth, 

present  (See  "  Present  worth  ") 


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